6409:
5600:
6404:{\displaystyle {\begin{aligned}{\boldsymbol {\pi }}^{(k)}&=\mathbf {x} \left(\mathbf {U\Sigma U} ^{-1}\right)\left(\mathbf {U\Sigma U} ^{-1}\right)\cdots \left(\mathbf {U\Sigma U} ^{-1}\right)\\&=\mathbf {xU\Sigma } ^{k}\mathbf {U} ^{-1}\\&=\left(a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\mathbf {u} _{n}^{\mathsf {T}}\right)\mathbf {U\Sigma } ^{k}\mathbf {U} ^{-1}\\&=a_{1}\lambda _{1}^{k}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\lambda _{2}^{k}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\lambda _{n}^{k}\mathbf {u} _{n}^{\mathsf {T}}&&u_{i}\bot u_{j}{\text{ for }}i\neq j\\&=\lambda _{1}^{k}\left\{a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{2}^{\mathsf {T}}+a_{3}\left({\frac {\lambda _{3}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{3}^{\mathsf {T}}+\cdots +a_{n}\left({\frac {\lambda _{n}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{n}^{\mathsf {T}}\right\}\end{aligned}}}
2857:
9481:, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description. For example, a thermodynamic state operates under a probability distribution that is difficult or expensive to acquire. Therefore, Markov Chain Monte Carlo method can be used to draw samples randomly from a black-box to approximate the probability distribution of attributes over a range of objects.
47:
9913:
20:
470:
2589:
14419:] Extensive, wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of state minimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment of Markov processes pp. 449ff. Discusses Z-transforms, D transforms in their context.
4565:
597:). Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term.
2246:
10619:
and position of the runners. Mark Pankin shows that Markov chain models can be used to evaluate runs created for both individual players as well as a team. He also discusses various kinds of strategies and play conditions: how Markov chain models have been used to analyze statistics for game situations such as
9384:, that every aperiodic and irreducible Markov chain is isomorphic to a Bernoulli scheme; thus, one might equally claim that Markov chains are a "special case" of Bernoulli schemes. The isomorphism generally requires a complicated recoding. The isomorphism theorem is even a bit stronger: it states that
9647:
towards a desired class of compounds such as drugs or natural products. As a molecule is grown, a fragment is selected from the nascent molecule as the "current" state. It is not aware of its past (that is, it is not aware of what is already bonded to it). It then transitions to the next state when a
9623:
A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions
644:
Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future. However, the statistical properties of the system's future can be predicted. In many applications, it is these statistical properties that are important.
628:
A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement. Formally, the steps are
10609:
Usually musical systems need to enforce specific control constraints on the finite-length sequences they generate, but control constraints are not compatible with Markov models, since they induce long-range dependencies that violate the Markov hypothesis of limited memory. In order to overcome this
787:
where, at each step, the position may change by +1 or â1 with equal probability. From any position there are two possible transitions, to the next or previous integer. The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example,
10618:
Markov chain models have been used in advanced baseball analysis since 1960, although their use is still rare. Each half-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered. During any at-bat, there are 24 possible combinations of number of outs
624:
describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not
9796:
by modeling texts in a natural language (such as
English) as generated by an ergodic Markov process, where each letter may depend statistically on previous letters. Such idealized models can capture many of the statistical regularities of systems. Even without describing the full structure of the
9754:
applications. Solar irradiance variability at any location over time is mainly a consequence of the deterministic variability of the sun's path across the sky dome and the variability in cloudiness. The variability of accessible solar irradiance on Earth's surface has been modeled using Markov
9628:
of molecules in solution in state A, each of which can undergo a chemical reaction to state B with a certain average rate. Perhaps the molecule is an enzyme, and the states refer to how it is folded. The state of any single enzyme follows a Markov chain, and since the molecules are essentially
4018:
of 1. If there is more than one unit eigenvector then a weighted sum of the corresponding stationary states is also a stationary state. But for a Markov chain one is usually more interested in a stationary state that is the limit of the sequence of distributions for some initial distribution.
669:
to hold. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption, which had been commonly
10183:
Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on a particular website can be modeled using first- or second-order Markov models and can be used to make predictions regarding future navigation and to personalize the web page for an
737:, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the KolmogorovâChapman equations. Other mathematicians who contributed significantly to the foundations of Markov processes include
1393:
is not possible. After the second draw, the third draw depends on which coins have so far been drawn, but no longer only on the coins that were drawn for the first state (since probabilistically important information has since been added to the scenario). In this way, the likelihood of the
780:, which are considered the most important and central stochastic processes in the theory of stochastic processes. These two processes are Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.
8603:
9372:
is a special case of a Markov chain where the transition probability matrix has identical rows, which means that the next state is independent of even the current state (in addition to being independent of the past states). A Bernoulli scheme with only two possible states is known as a
4432:
8881:
9655:
may be modeled using Markov chains. Based on the reactivity ratios of the monomers that make up the growing polymer chain, the chain's composition may be calculated (for example, whether monomers tend to add in alternating fashion or in long runs of the same monomer). Due to
3623:
9248:
9611:
8399:
is the time, starting in a given set of states until the chain arrives in a given state or set of states. The distribution of such a time period has a phase type distribution. The simplest such distribution is that of a single exponentially distributed transition.
2195:
4978:
whose each row sums to 1. So it needs any nĂn independent linear equations of the (nĂn+n) equations to solve for the nĂn variables. In this example, the n equations from âQ multiplied by the right-most column of (P-In)â have been replaced by the n stochastic
4438:
9311:
Markov models are used to model changing systems. There are 4 main types of models, that generalize Markov chains depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made:
12800:
Kopp, V. S.; Kaganer, V. M.; Schwarzkopf, J.; Waidick, F.; Remmele, T.; Kwasniewski, A.; Schmidbauer, M. (2011). "X-ray diffraction from nonperiodic layered structures with correlations: Analytical calculation and experiment on mixed
Aurivillius films".
1733:
497:"): it is a process for which predictions can be made regarding future outcomes based solely on its present state andâmost importantlyâsuch predictions are just as good as the ones that could be made knowing the process's full history. This means that,
5540:
729:'s work on Einstein's model of Brownian movement. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov's work,
5336:
584:
Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a
7945:
is the set of all states for the Markov chain. Let the sigma-algebra on the probability space be generated by the cylinder sets. Let the probability measure be generated by the stationary distribution, and the Markov chain transition. Let
791:
A series of independent states (for example, a series of coin flips) satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next state depends on the current
9640:, can be viewed as a Markov chain, where at each time step the reaction proceeds in some direction. While Michaelis-Menten is fairly straightforward, far more complicated reaction networks can also be modeled with Markov chains.
2584:{\displaystyle {\begin{aligned}{}&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{1}=x_{1})\\=&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{n-m}=x_{n-m}){\text{ for }}n>m\end{aligned}}}
11665:
Kendall, D. G.; Batchelor, G. K.; Bingham, N. H.; Hayman, W. K.; Hyland, J. M. E.; Lorentz, G. G.; Moffatt, H. K.; Parry, W.; Razborov, A. A.; Robinson, C. A.; Whittle, P. (1990). "Andrei
Nikolaevich Kolmogorov (1903â1987)".
11485:
Kendall, D. G.; Batchelor, G. K.; Bingham, N. H.; Hayman, W. K.; Hyland, J. M. E.; Lorentz, G. G.; Moffatt, H. K.; Parry, W.; Razborov, A. A.; Robinson, C. A.; Whittle, P. (1990). "Andrei
Nikolaevich Kolmogorov (1903â1987)".
516:
with a countable state space (thus regardless of the nature of time), but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).
801:
Suppose that there is a coin purse containing five quarters (each worth 25Âą), five dimes (each worth 10Âą), and five nickels (each worth 5Âą), and one by one, coins are randomly drawn from the purse and are set on a table. If
4763:
529:
and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v. continuous time:
656:
studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906. Markov
Processes in continuous time were discovered long before his work in the early 20th century in the form of the
1248:
possible states, where each state represents the number of coins of each type (from 0 to 5) that are on the table. (Not all of these states are reachable within 6 draws.) Suppose that the first draw results in state
8470:
4573:
Because there are a number of different special cases to consider, the process of finding this limit if it exists can be a lengthy task. However, there are many techniques that can assist in finding this limit. Let
5064:
4208:
1972:
5345:
is a row stochastic matrix, its largest left eigenvalue is 1. If there is a unique stationary distribution, then the largest eigenvalue and the corresponding eigenvector is unique too (because there is no other
4324:
3158:
8757:
2767:
6975:
767:
problem are examples of Markov processes. Some variations of these processes were studied hundreds of years earlier in the context of independent variables. Two important examples of Markov processes are the
10671:. The Markov chain forecasting models utilize a variety of settings, from discretizing the time series, to hidden Markov models combined with wavelets, and the Markov chain mixture distribution model (MCM).
10123:
9969:
8958:
4640:
6768:
6543:
724:
developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. Kolmogorov was partly inspired by Louis
Bachelier's 1900 work on fluctuations in the stock market as well as
709:
in 1873, preceding the work of Markov. After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by
7019:
otherwise. Periodicity, transience, recurrence and positive and null recurrence are class properties â that is, if one state has the property then all states in its communicating class have the property.
6627:
with each other if both are reachable from one another by a sequence of transitions that have positive probability. This is an equivalence relation which yields a set of communicating classes. A class is
1988:
10349:
for each note is constructed, completing a transition probability matrix (see below). An algorithm is constructed to produce output note values based on the transition matrix weightings, which could be
10317:", for example, are represented exactly by Markov chains. At each turn, the player starts in a given state (on a given square) and from there has fixed odds of moving to certain other states (squares).
3384:
788:
the transition probabilities from 5 to 4 and 5 to 6 are both 0.5, and all other transition probabilities from 5 are 0. These probabilities are independent of whether the system was previously in 4 or 6.
1824:
620:
The changes of state of the system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, a
7529:
9133:
7039:
Since periodicity is a class property, if a Markov chain is irreducible, then all its states have the same period. In particular, if one state is aperiodic, then the whole Markov chain is aperiodic.
4966:
10285:. In current research, it is common to use a Markov chain to model how once a country reaches a specific level of economic development, the configuration of structural factors, such as size of the
5209:
10212:
Markov chains are used in finance and economics to model a variety of different phenomena, including the distribution of income, the size distribution of firms, asset prices and market crashes.
9865:
initiated the subject in 1917. This makes them critical for optimizing the performance of telecommunications networks, where messages must often compete for limited resources (such as bandwidth).
10599:
th-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense of
8358:
8056:
2205:. Every stationary chain can be proved to be time-homogeneous by Bayes' rule.A necessary and sufficient condition for a time-homogeneous Markov chain to be stationary is that the distribution of
8475:
5605:
2251:
3958:
3826:
4316:
4264:
604:
of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space. However, many applications of Markov chains employ finite or
12327:
10192:
Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called
9500:
12765:
Kutchukian, Peter S.; Lou, David; Shakhnovich, Eugene I. (2009-06-15). "FOG: Fragment
Optimized Growth Algorithm for the de Novo Generation of Molecules Occupying Druglike Chemical Space".
9624:
of the chain. Markov chains and continuous-time Markov processes are useful in chemistry when physical systems closely approximate the Markov property. For example, imagine a large number
8091:
7923:
4560:{\displaystyle {\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}
11343:
Guttorp, Peter; Thorarinsdottir, Thordis L. (2012). "What
Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes".
4681:
911:
608:
state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations (see
6600:
512:
or a discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either
10240:
and Adlai J. Fisher, which builds upon the convenience of earlier regime-switching models. It uses an arbitrarily large Markov chain to drive the level of volatility of asset returns.
8237:
In some cases, apparently non-Markovian processes may still have
Markovian representations, constructed by expanding the concept of the "current" and "future" states. For example, let
3022:
593:
without explicit mention. In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (see
4102:
641:
for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps.
10158:
8675:
3894:
8692:
states that the necessary and sufficient condition for a process to be reversible is that the product of transition rates around a closed loop must be the same in both directions.
1543:
1246:
10606:
Markov chains can be used structurally, as in
Xenakis's Analogique A and B. Markov chains are also used in systems which use a Markov model to react interactively to music input.
10232:(1989), who used a Markov chain to model switches between periods of high and low GDP growth (or, alternatively, economic expansions and recessions). A more recent example is the
7976:
5443:
2845:
are chosen such that each row of the transition rate matrix sums to zero, while the row-sums of a probability transition matrix in a (discrete) Markov chain are all equal to one.
14178:
Munkhammar, J.; van der Meer, D.W.; Widén, J. (2019). "Probabilistic forecasting of high-resolution clear-sky index time-series using a Markov-chain mixture distribution model".
9043:
7851:
4049:
7659:
7382:
5435:
1488:
1017:
14596:
Original paper by A.A Markov (1913): An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains (translated from Russian)
12960:
Aguiar, R. J.; Collares-Pereira, M.; Conde, J. P. (1988). "Simple procedure for generating sequences of daily radiation values using a library of Markov transition matrices".
9120:
7457:
4004:
3698:
1210:
could be defined to represent the state where there is one quarter, zero dimes, and five nickels on the table after 6 one-by-one draws. This new model could be represented by
7765:
954:
12730:
Kutchukian, Peter; Lou, David; Shakhnovich, Eugene (2009). "FOG: Fragment Optimized Growth Algorithm for the de Novo Generation of Molecules occupying Druglike Chemical".
8196:
7097:
5220:
4055:
and its eigenvectors have their relative proportions preserved. Since the components of π are positive and the constraint that their sum is unity can be rewritten as
3188:
14570:
8170:
7616:
1437:
1391:
1292:
1208:
13976:
8450:
7718:
2973:
9086:
9066:
9008:
2860:
The continuous time Markov chain is characterized by the transition rates, the derivatives with respect to time of the transition probabilities between states i and j.
371:
of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs
10178:
8216:
7943:
7578:
2934:
2659:
2626:
864:
7172:
It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state. More generally, a Markov chain is ergodic if there is a number
6801:
3226:
10068:
7686:
7322:
7255:
7167:
7013:
2889:
2230:
1346:
1319:
1159:
1128:
1098:
1071:
1044:
981:
827:
14668:
983:, but the earlier values as well, then we can determine which coins have been drawn, and we know that the next coin will not be a nickel; so we can determine that
6827:
9272:
Another discrete-time process that may be derived from a continuous-time Markov chain is a ÎŽ-skeleton—the (discrete-time) Markov chain formed by observing
14224:
A. A. Markov (1971). "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard.
10041:
10021:
10001:
8125:
7871:
7785:
7552:
7477:
7405:
7342:
7295:
7228:
7140:
4266:
is found, then the stationary distribution of the Markov chain in question can be easily determined for any starting distribution, as will be explained below.
1046:
we might guess that we had drawn four dimes and two nickels, in which case it would certainly be possible to draw another nickel next. Thus, our guesses about
9454:
9450:
9739:", arranging these chains in several recursive layers ("wafering") and producing more efficient test setsâsamplesâas a replacement for exhaustive testing.
1857:
7195:
Some authors call any irreducible, positive recurrent Markov chains ergodic, even periodic ones. In fact, merely irreducible Markov chains correspond to
7187:
A Markov chain with more than one state and just one out-going transition per state is either not irreducible or not aperiodic, hence cannot be ergodic.
3039:
15203:
338:
12131:
4696:
13329:
15027:
9824:
Markov chains are also the basis for hidden Markov models, which are an important tool in such diverse fields as telephone networks (which use the
8100:
Markov chains with finite state spaces have a unique stationary distribution, the above construction is unambiguous for irreducible Markov chains.
4842:
and substitutes each of its elements by one, and on the other one substitutes the corresponding element (the one in the same column) in the vector
8221:
The terminology is inconsistent. Given a Markov chain with a stationary distribution that is strictly positive on all states, the Markov chain is
9442:
8598:{\displaystyle {\begin{aligned}k_{i}^{A}=0&{\text{ for }}i\in A\\-\sum _{j\in S}q_{ij}k_{j}^{A}=1&{\text{ for }}i\notin A.\end{aligned}}}
15630:
14623:
14320:. Grundlehren der mathematischen Wissenschaften. Vol. I (121). Translated by Fabius, Jaap; Greenberg, Vida Lazarus; Maitra, Ashok Prasad;
14258:
11979:
Schmitt, Florian; Rothlauf, Franz (2001). "On the Importance of the Second Largest Eigenvalue on the Convergence Rate of Genetic Algorithms".
10243:
Dynamic macroeconomics makes heavy use of Markov chains. An example is using Markov chains to exogenously model prices of equity (stock) in a
15160:
15140:
10265:
arguments, where current structural configurations condition future outcomes. An example is the reformulation of the idea, originally due to
13652:
Calvet, Laurent; Adlai Fisher (2004). "How to Forecast long-run volatility: regime-switching and the estimation of multifractal processes".
12995:
Ngoko, B. O.; Sugihara, H.; Funaki, T. (2014). "Synthetic generation of high temporal resolution solar radiation data using Markov models".
5031:
4427:{\displaystyle \mathbf {P} ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \mathbf {P} ^{2k}=I\qquad \mathbf {P} ^{2k+1}=\mathbf {P} }
14475:. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973)
9349:
8876:{\displaystyle s_{ij}={\begin{cases}{\frac {q_{ij}}{\sum _{k\neq i}q_{ik}}}&{\text{if }}i\neq j\\0&{\text{otherwise}}.\end{cases}}}
7180:. In case of a fully connected transition matrix, where all transitions have a non-zero probability, this condition is fulfilled with
4157:
14233:
Markov, A. A. (2006). "An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains".
11300:
Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: The early years, 1880â1970".
15544:
14407:
13908:
13690:
6599:
Considering a collection of Markov chains whose evolution takes in account the state of other Markov chains, is related to the notion of
7122:
If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. Equivalently, there exists some integer
2672:
13718:
6874:
9735:
Several theorists have proposed the idea of the Markov chain statistical test (MCST), a method of conjoining Markov chains to form a "
9465:
Markov chains have been employed in a wide range of topics across the natural and social sciences, and in technological applications.
7042:
If a finite Markov chain is irreducible, then all states are positive recurrent, and it has a unique stationary distribution given by
15461:
12192:
10073:
9919:
9836:
8896:
6574:
670:
regarded as a requirement for such mathematical laws to hold. Markov later used Markov chains to study the distribution of vowels in
552:
14302:
6679:
6479:
4589:
3618:{\displaystyle \Pr(X_{t_{n+1}}=i_{n+1}\mid X_{t_{0}}=i_{0},X_{t_{1}}=i_{1},\ldots ,X_{t_{n}}=i_{n})=p_{i_{n}i_{n+1}}(t_{n+1}-t_{n})}
15145:
10228:
and random walk models were popular in the literature of the 1960s. Regime-switching models of business cycles were popularized by
612:). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise.
15471:
15155:
13308:
9243:{\displaystyle \pi ={-\varphi (\operatorname {diag} (Q))^{-1} \over \left\|\varphi (\operatorname {diag} (Q))^{-1}\right\|_{1}}.}
11956:
1742:
15513:
15410:
14567:
9784:
7482:
331:
23:
A diagram representing a two-state Markov process. The numbers are the probability of changing from one state to another state.
14595:
13980:
12416:. Probability and its applications (2. ed., ed.). New York, NY Berlin Heidelberg: Springer. Proposition 8.6 (page 145).
10224:
was the first to observe that stock prices followed a random walk. The random walk was later seen as evidence in favor of the
6591:
The use of Markov chains in Markov chain Monte Carlo methods covers cases where the process follows a continuous state space.
4871:
15700:
15690:
15536:
15228:
15213:
14534:
14480:
14403:
14370:
14341:
14030:
13851:
13370:
12615:
12421:
12363:
12311:
11940:
11868:
11824:
11797:
11737:
11707:
11649:
11587:
11560:
11464:
11327:
11284:
11222:
11190:
11055:
11028:
11001:
10974:
10947:
10850:
10823:
10796:
5165:
5004:
th row or column is otherwise filled with 0's, then that row or column will remain unchanged in all of the subsequent powers
1537:, namely that the probability of moving to the next state depends only on the present state and not on the previous states:
15600:
15564:
9723:
5110:
be the matrix of eigenvectors (each normalized to having an L2 norm equal to 1) where each column is a left eigenvector of
638:
9606:{\displaystyle {\ce {{E}+{\underset {Substrate \atop binding}{S<=>E}}{\overset {Catalytic \atop step}{S->E}}+P}}}
8259:
7981:
15868:
15605:
9618:. The enzyme (E) binds a substrate (S) and produces a product (P). Each reaction is a state transition in a Markov chain.
8701:
7887:
6584:
Many results for Markov chains with finite state space can be generalized to chains with uncountable state space through
3909:
3745:
15517:
14715:
14616:
13574:
Hamilton, James (1989). "A new approach to the economic analysis of nonstationary time series and the business cycle".
11770:
679:
13032:"Stochastic generation of synthetic minutely irradiance time series derived from mean hourly weather observation data"
6559:
is the dominant term. The smaller the ratio is, the faster the convergence is. Random noise in the state distribution
4276:
4224:
2811:
with dimensions equal to that of the state space and initial probability distribution defined on the state space. For
718:
became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.
15670:
14519:
14512:
Performance and reliability analysis of computer systems: an example-based approach using the SHARPE software package
14497:
14466:
14461:
E. Nummelin. "General irreducible Markov chains and non-negative operators". Cambridge University Press, 1984, 2004.
14456:
14298:
14283:
14265:
13930:
13141:
Munkhammar, J.; Widén, J. (2018). "A Markov-chain probability distribution mixture approach to the clear-sky index".
12648:
12479:
12253:
12172:
12141:
11904:
11141:
11120:
11099:
11079:
10691:
9648:
fragment is attached to it. The transition probabilities are trained on databases of authentic classes of compounds.
3642:
2190:{\displaystyle \Pr(X_{0}=x_{0},X_{1}=x_{1},\ldots ,X_{k}=x_{k})=\Pr(X_{n}=x_{0},X_{n+1}=x_{1},\ldots ,X_{n+k}=x_{k})}
502:
324:
312:
271:
8061:
7893:
5091:
linearly independent eigenvectors, speed of convergence is elaborated as follows. (For non-diagonalizable, that is,
15715:
15521:
15505:
15420:
15248:
15218:
14640:
12631:
Anderson, David F.; Kurtz, Thomas G. (2011), "Continuous Time Markov Chain Models for Chemical Reaction Networks",
3855:
is a (row) vector, whose entries are non-negative and sum to 1, is unchanged by the operation of transition matrix
661:. Markov was interested in studying an extension of independent random sequences, motivated by a disagreement with
15909:
15620:
15585:
15554:
15549:
14985:
14902:
10681:
10220:
and co-author Charles Bonini used a Markov chain model to derive a stationary Yule distribution of firm sizes.
4650:
920:
To see why this is the case, suppose that in the first six draws, all five nickels and a quarter are drawn. Thus
869:
734:
513:
202:
138:
7324:
are positive. The exponent is purely a graph-theoretic property, since it depends only on whether each entry of
693:
with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by
15559:
15188:
15183:
14990:
14887:
13995:
13875:
13071:
Munkhammar, J.; Widén, J. (2018). "An N-state Markov-chain mixture distribution model of the clear-sky index".
12905:"Comparison of Parameter Estimation Methods in Stochastic Chemical Kinetic Models: Examples in Systems Biology"
10909:
9904:(job service times are exponentially distributed) and describe completed services (departures) from the queue.
2978:
730:
250:
111:
1728:{\displaystyle \Pr(X_{n+1}=x\mid X_{1}=x_{1},X_{2}=x_{2},\ldots ,X_{n}=x_{n})=\Pr(X_{n+1}=x\mid X_{n}=x_{n}),}
15873:
15650:
15486:
15385:
15370:
14909:
14782:
14698:
14609:
14560:
13887:
10751:
10233:
9637:
9388:
5535:{\displaystyle \mathbf {x} ^{\mathsf {T}}=\sum _{i=1}^{n}a_{i}\mathbf {u} _{i},\qquad a_{i}\in \mathbb {R} .}
571:
15645:
15525:
10128:
9767:
8625:
4058:
3865:
711:
15655:
14431:(1st ed.). Englewood Cliffs, NJ: Prentice-Hall, Inc. Library of Congress Card Catalog Number 59-12841.
9381:
6608:
4218:
2779:
1213:
388:
15660:
15296:
7949:
4818:(see the definition above). It is sometimes sufficient to use the matrix equation above and the fact that
15258:
14842:
14787:
14703:
14555:
14417:(1st ed.). New York, NY: John Wiley and Sons, Inc. Library of Congress Card Catalog Number 67-25924.
14109:
13326:
10761:
10640:
10225:
9016:
7794:
7176:
such that any state can be reached from any other state in any number of steps less or equal to a number
6604:
4025:
15590:
7624:
7347:
15595:
15580:
15223:
15193:
14760:
14658:
14585:
14217:
A. A. Markov (1906) "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga".
10696:
5553:
from right and continue this operation with the results, in the end we get the stationary distribution
5408:
4971:
1504:
1442:
986:
380:
106:
14050:
10663:
Markov chains have been used for forecasting in several areas: for example, price trends, wind power,
10655:, and Academias Neutronium). Several open-source text generation libraries using Markov chains exist.
9092:
7414:
15894:
15675:
15476:
15390:
15375:
15306:
14882:
14765:
14663:
10873:
10716:
10706:
9615:
8689:
5331:{\displaystyle 1=|\lambda _{1}|>|\lambda _{2}|\geq |\lambda _{3}|\geq \cdots \geq |\lambda _{n}|.}
3966:
754:
666:
222:
13772:
13666:
13588:
9755:
chains, also including modeling the two states of clear and cloudiness as a two-state Markov chain.
8782:
7387:
There are several combinatorial results about the exponent when there are finitely many states. Let
15509:
15395:
14897:
14872:
14817:
14309:
13448:
13415:
13387:
13313:
13215:
Thomsen, Samuel W. (2009), "Some evidence concerning the genesis of Shannon's information theory",
11989:
11310:
10756:
10701:
10686:
10193:
9705:
9643:
An algorithm based on a Markov chain was also used to focus the fragment-based growth of chemicals
7726:
3288:
923:
411:
407:
406:
of real-world processes. They provide the basis for general stochastic simulation methods known as
281:
276:
165:
150:
14550:
12220:
8175:
7045:
4846:, and next left-multiplies this latter vector by the inverse of transformed former matrix to find
3163:
15810:
15800:
15615:
15491:
15273:
15198:
15012:
14877:
14733:
14688:
13625:
12098:
10878:
10711:
10326:
9687:
9438:
9400:
9344:
8740:
8720:. Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as a
8390:
8130:
7874:
7583:
6658:
4139:
If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution
3840:
3651:
1736:
1397:
1351:
1252:
1168:
498:
260:
131:
13303:
12049:
Franzke, Brandon; Kosko, Bart (1 October 2011). "Noise can speed convergence in Markov chains".
9916:
A state diagram that represents the PageRank algorithm with a transitional probability of M, or
15899:
15752:
15680:
15105:
15095:
14939:
13767:
13661:
13583:
13443:
13410:
11984:
11305:
10201:
9818:
8423:
7691:
2939:
2805:
155:
14147:
13905:
13701:
13360:
13243:"An alignment-free method to find and visualise rearrangements between pairs of DNA sequences"
9071:
9051:
8993:
6603:. This corresponds to the situation when the state space has a (Cartesian-) product form. See
637:, and the random process is a mapping of these to states. The Markov property states that the
15904:
15775:
15757:
15737:
15732:
15451:
15283:
15263:
15110:
15053:
14892:
14802:
13815:
K McAlpine; E Miranda; S Hoggar (1999). "Making Music with Algorithms: A Case-Study System".
11697:
10886:
10253:
produce annual tables of the transition probabilities for bonds of different credit ratings.
10163:
9478:
9422:
8201:
7928:
7557:
3732:
2906:
2631:
2598:
836:
658:
296:
255:
160:
126:
14590:
14425:
13729:
11639:
6780:
3201:
15850:
15805:
15795:
15481:
15456:
15425:
15405:
15243:
15165:
15150:
15017:
14187:
13759:
13402:
13254:
13150:
13115:
13080:
13043:
13004:
12969:
12916:
12857:
12810:
12686:
12058:
11612:
10741:
10664:
10278:
10250:
10046:
8375:
7788:
7664:
7300:
7233:
7145:
6991:
3720:
2867:
2208:
1324:
1297:
1137:
1106:
1076:
1049:
1022:
1019:
with probability 1. But if we do not know the earlier values, then based only on the value
959:
805:
706:
286:
180:
73:
2856:
8:
15845:
15685:
15610:
15415:
15175:
15085:
14975:
14321:
13299:
12521:
10746:
10731:
10244:
10003:
in the stationary distribution on the following Markov chain on all (known) webpages. If
9862:
9814:
9793:
9763:
9701:
9660:, second-order Markov effects may also play a role in the growth of some polymer chains.
9334:
6806:
415:
245:
187:
175:
170:
14490:
Probability and Statistics with Reliability, Queueing, and Computer Science Applications
14191:
13763:
13437:
13406:
13258:
13154:
13119:
13084:
13047:
13008:
12973:
12920:
12861:
12814:
12690:
12062:
11616:
11408:
Seneta, E. (1998). "I.J. Bienaymé : Criticality, Inequality, and Internationalization".
4974:
in nĂn variables. And there are n more linear equations from the fact that Q is a right
1103:
However, it is possible to model this scenario as a Markov process. Instead of defining
15815:
15780:
15695:
15665:
15496:
15435:
15430:
15253:
15090:
14755:
14693:
14632:
13792:
13747:
13601:
13480:
13275:
13242:
13177:
12937:
12904:
12880:
12845:
12707:
12676:
12664:
12573:
12548:
12500:
12025:
11425:
11390:
10346:
10310:
10213:
10197:
10026:
10006:
9986:
9789:
9663:
Similarly, it has been suggested that the crystallization and growth of some epitaxial
9089:
8110:
7856:
7770:
7537:
7462:
7390:
7327:
7280:
7213:
7125:
7119:
is ergodic if it is recurrent, has a period of 1, and has finite mean recurrence time.
5157:
5096:
4810:. Multiplying together stochastic matrices always yields another stochastic matrix, so
486:
435:
376:
360:
232:
121:
61:
38:
14568:
Markov Chains chapter in American Mathematical Society's introductory probability book
14422:
14351:
14314:
13638:
13362:
Handbook of Research on Modern Cryptographic Solutions for Computer and Cyber Security
13127:
12353:
12301:
10842:
Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition
10345:. In a first-order chain, the states of the system become note or pitch values, and a
8107:, a measure-preserving dynamical system is called "ergodic" iff any measurable subset
4108:
of π with a vector whose components are all 1 is unity and that π lies on a
715:
15835:
15048:
14965:
14934:
14827:
14807:
14797:
14653:
14648:
14530:
14515:
14507:, vol. 36, no. 4, pp. 52â57, ACM SIGMETRICS Performance Evaluation Review, 2009.
14493:
14485:
14476:
14462:
14452:
14399:
14366:
14337:
14294:
14279:
14261:
14026:
14019:
13871:
13847:
13797:
13366:
13280:
13197:
12981:
12942:
12885:
12826:
12782:
12747:
12712:
12644:
12611:
12578:
12475:
12417:
12397:
12380:
12359:
12307:
12249:
12168:
12137:
12112:
12093:
12074:
11936:
11900:
11864:
11820:
11793:
11766:
11733:
11703:
11645:
11583:
11556:
11460:
11356:
11323:
11280:
11218:
11186:
11137:
11116:
11095:
11075:
11051:
11024:
10997:
10970:
10943:
10915:
10905:
10867:
10846:
10819:
10792:
10648:
10298:
10237:
10229:
9825:
9806:
9374:
5073:
4975:
4815:
3195:
764:
721:
698:
675:
621:
447:
443:
403:
291:
197:
96:
15640:
15291:
12438:
11762:
11182:
7688:
has positive diagonal entries, which by previous proposition means its exponent is
5396:
row vector that represents a valid probability distribution; since the eigenvectors
4838:
unknowns, so it is computationally easier if on the one hand one selects one row in
686:
15855:
15742:
15625:
15501:
15238:
14995:
14970:
14919:
14770:
14723:
14358:
14329:
14242:
14195:
14121:
14088:
13824:
13787:
13777:
13671:
13634:
13593:
13539:
13534:
13526:
13472:
13420:
13270:
13262:
13224:
13189:
13158:
13123:
13088:
13051:
13012:
12977:
12932:
12924:
12875:
12865:
12818:
12774:
12739:
12702:
12694:
12636:
12603:
12568:
12560:
12467:
12392:
12127:
12107:
12066:
12017:
11928:
11892:
11758:
11675:
11620:
11529:
11495:
11452:
11417:
11382:
11352:
11315:
11252:
10668:
10644:
10217:
9810:
9802:
9798:
9747:
9704:, where Markov chains are in particular a central tool in the theoretical study of
9426:
9421:. A Markov matrix that is compatible with the adjacency matrix can then provide a
9406:
9369:
9363:
8678:
5092:
4109:
3638:
116:
46:
14847:
14199:
13162:
13092:
13056:
13031:
13016:
12276:
15820:
15720:
15705:
15466:
15400:
15078:
15022:
15005:
14750:
14574:
14444:
14325:
13917:
Proceedings of the 22nd International Joint Conference on Artificial Intelligence
13912:
13333:
12870:
12597:
12471:
12459:
11896:
11884:
11858:
11841:
11814:
11787:
11727:
11577:
11550:
11456:
11274:
11212:
11045:
11018:
10991:
10964:
10937:
10840:
10813:
10786:
10736:
10726:
10721:
10643:
given a sample document. Markov processes are used in a variety of recreational "
10620:
10603:
structure, rather than the 'aimless wandering' produced by a first-order system.
10314:
10221:
9885:
9858:
9852:
9717:
9434:
8976:
8968:
4778:
4318:
does not exist while the stationary distribution does, as shown by this example:
3846:
3704:
3191:
1534:
1510:
777:
773:
634:
490:
384:
192:
143:
15635:
14867:
13228:
12640:
12508:
9714:, where Markov chains have been used, e.g., to simulate the mammalian neocortex.
9629:
independent of each other, the number of molecules in state A or B at a time is
9391:
is isomorphic to a Bernoulli scheme; the Markov chain is just one such example.
4147:
converges to a rank-one matrix in which each row is the stationary distribution
15825:
15790:
15710:
15316:
15063:
14980:
14949:
14944:
14924:
14914:
14857:
14832:
14812:
14777:
14745:
14728:
14440:
14093:
14076:
13193:
12070:
10652:
10600:
10591:
A second-order Markov chain can be introduced by considering the current state
10338:
10262:
9829:
9779:
9736:
9683:
9657:
9474:
9430:
8453:
8104:
7886:
If a Markov chain has a stationary distribution, then it can be converted to a
7200:
4758:{\displaystyle \mathbf {Q} (\mathbf {P} -\mathbf {I} _{n})=\mathbf {0} _{n,n},}
2661:
which has the 'classical' Markov property by taking as state space the ordered
783:
A famous Markov chain is the so-called "drunkard's walk", a random walk on the
769:
738:
726:
702:
694:
662:
575:
494:
207:
14852:
14579:
14423:
Kemeny, John G.; Hazleton Mirkil; J. Laurie Snell; Gerald L. Thompson (1959).
14362:
14333:
14246:
13358:
13346:
12822:
12607:
12602:. Lecture Notes in Physics. Vol. 788. Springer-Verlag Berlin Heidelberg.
12564:
11932:
11319:
10309:
Markov chains can be used to model many games of chance. The children's games
9675:
Markov chains are used in various areas of biology. Notable examples include:
15888:
15727:
15268:
15100:
15058:
15000:
14822:
14738:
14678:
13675:
13201:
12272:
12216:
11534:
11517:
11370:
10624:
10342:
10294:
10293:
mobilization, etc., will generate a higher probability of transitioning from
9912:
9840:
9679:
8980:
2828:
are non-negative and describe the rate of the process transitions from state
1839:
742:
671:
653:
605:
474:
396:
80:
13828:
13782:
13498:
Simon, Herbert; C Bonini (1958). "The size distribution of business firms".
13106:
Morf, H. (1998). "The stochastic two-state solar irradiance model (STSIM)".
10919:
9868:
Numerous queueing models use continuous-time Markov chains. For example, an
8374:
An example of a non-Markovian process with a Markovian representation is an
15785:
15747:
15301:
15233:
15122:
15117:
14929:
14862:
14837:
13801:
13620:
13284:
12946:
12889:
12830:
12786:
12751:
12716:
12582:
12078:
11679:
11499:
10286:
9711:
9664:
9446:
9410:
9306:
8722:
6585:
690:
594:
556:
307:
101:
13934:
11176:
11174:
10902:
Stochastic differential equations : an introduction with applications
8990:
To find the stationary probability distribution vector, we must next find
8681:
this process has the same stationary distribution as the forward process.
4115:
15830:
15365:
15349:
15344:
15339:
15329:
15132:
15073:
15068:
15032:
14792:
14683:
13956:
12665:"Correlation analysis of enzymatic reaction of a single protein molecule"
12188:
11444:
10272:
9869:
9751:
9485:
9266:
8726:. Each element of the one-step transition probability matrix of the EMC,
8378:
7344:
is zero or positive, and therefore can be found on a directed graph with
5103:
and proceed with a bit more involved set of arguments in a similar way.)
4799:
4105:
3900:
784:
760:
601:
526:
509:
379:
sequence, in which the chain moves state at discrete time steps, gives a
368:
227:
68:
56:
11552:
Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters
11548:
11256:
4217:
is the column vector with all entries equal to 1. This is stated by the
15840:
15380:
15324:
15208:
15161:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
14601:
14271:
14219:
Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete
13605:
13530:
13484:
13438:
Page, Lawrence; Brin, Sergey; Motwani, Rajeev; Winograd, Terry (1999).
12729:
12698:
12029:
12005:
11429:
11394:
10282:
9691:
7196:
4015:
3716:
85:
31:
14051:"Forecasting oil price trends using wavelets and hidden Markov models"
14049:
de Souza e Silva, E.G.; Legey, L.F.L.; de Souza e Silva, E.A. (2010).
13424:
13266:
12928:
12778:
12743:
11624:
10196:(MCMC). In recent years this has revolutionized the practicability of
9797:
system perfectly, such signal models can make possible very effective
9292:(2ÎŽ), ... give the sequence of states visited by the ÎŽ-skeleton.
6848:, there is a non-zero probability that the chain will never return to
5059:{\displaystyle {\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P} ,}
508:
A Markov chain is a type of Markov process that has either a discrete
15334:
14391:
in 1963 and translated to English with the assistance of the author.)
14388:
14125:
14048:
13745:
11518:"Half a Century with Probability Theory: Some Personal Recollections"
10628:
10290:
10266:
9872:
is a CTMC on the non-negative integers where upward transitions from
9652:
9644:
4203:{\displaystyle \lim _{k\to \infty }\mathbf {P} ^{k}=\mathbf {1} \pi }
1967:{\displaystyle \Pr(X_{n+1}=x\mid X_{n}=y)=\Pr(X_{n}=x\mid X_{n-1}=y)}
427:
423:
13597:
13476:
13178:"A Systematic Review of Hidden Markov Models and Their Applications"
12021:
11789:
The Wonderful world of stochastics: a tribute to Elliott W. Montroll
11421:
11386:
9857:
Markov chains are the basis for the analytical treatment of queues (
9720:, for instance with the modeling of viral infection of single cells.
4120:
If the Markov chain is time-homogeneous, then the transition matrix
461:
are used to describe something that is related to a Markov process.
10595:
also the previous state, as indicated in the second table. Higher,
10330:
10216:
built a Markov chain model of the distribution of income in 1953.
9976:
7115:
if it is aperiodic and positive recurrent. In other words, a state
6632:
if the probability of leaving the class is zero. A Markov chain is
6565:
can also speed up this convergence to the stationary distribution.
5023:
3153:{\displaystyle \Pr(X(t+h)=j\mid X(t)=i)=\delta _{ij}+q_{ij}h+o(h),}
2891:
be the random variable describing the state of the process at time
630:
501:
on the present state of the system, its future and past states are
364:
212:
14077:"Markov chain modeling for very-short-term wind power forecasting"
13891:
13359:
Gupta, Brij; Agrawal, Dharma P.; Yamaguchi, Shingo (16 May 2016).
12681:
12547:
van Ravenzwaaij, Don; Cassey, Pete; Brown, Scott D. (2016-03-11).
10811:
9983:
is defined by a Markov chain. It is the probability to be at page
8367:
has the Markov property, then it is a Markovian representation of
4865:
with its right-most column replaced with all 1's. If exists then
3719:, the transition probability distribution can be represented by a
2762:{\displaystyle Y_{n}=\left(X_{n},X_{n-1},\ldots ,X_{n-m+1}\right)}
549:(discrete-time) Markov chain on a countable or finite state space
13240:
12355:
Non-negative matrices; an introduction to theory and applications
12303:
Non-negative matrices; an introduction to theory and applications
11981:
Proceedings of the 14th Symposium on Reliable Distributed Systems
11856:
8709:
6970:{\displaystyle M_{i}=E=\sum _{n=1}^{\infty }n\cdot f_{ii}^{(n)}.}
439:
431:
419:
11641:
Continuous-Time Markov Chains: An Applications-Oriented Approach
9839:
lossless data compression algorithm combines Markov chains with
589:, but a few authors use the term "Markov process" to refer to a
19:
12439:"Smoothing of noisy AR signals using an adaptive Kalman filter"
11549:
Marc Barbut; Bernard Locker; Laurent Mazliak (23 August 2016).
11214:
Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues
10899:
10334:
10118:{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}
9980:
9964:{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}
9695:
8953:{\displaystyle S=I-\left(\operatorname {diag} (Q)\right)^{-1}Q}
4635:{\textstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.}
829:
represents the total value of the coins set on the table after
392:
13698:
Department of Finance, the Anderson School of Management, UCLA
13030:
Bright, J. M.; Smith, C. I.; Taylor, P. G.; Crook, R. (2015).
12844:
George, Dileep; Hawkins, Jeff (2009). Friston, Karl J. (ed.).
12133:
Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis
11480:
11478:
11476:
10838:
9667:
oxide materials can be accurately described by Markov chains.
9429:
are isomorphic to topological Markov chains; examples include
6763:{\displaystyle k=\gcd\{n>0:\Pr(X_{n}=i\mid X_{0}=i)>0\}}
6538:{\displaystyle |\lambda _{2}|\geq \cdots \geq |\lambda _{n}|,}
5352:
which solves the stationary distribution equation above). Let
3847:
Stationary distribution relation to eigenvectors and simplices
14305:. Second edition to appear, Cambridge University Press, 2009.
14257:. Original edition published by Addison-Wesley; reprinted by
14074:
12799:
12352:
Seneta, E. (Eugene) (1973). "2.4. Combinatorial properties".
11725:
11664:
11484:
11170:
11168:
11166:
11164:
11162:
10989:
10355:
8225:
iff its corresponding measure-preserving dynamical system is
7459:. The only case where it is an equality is when the graph of
12549:"A simple introduction to Markov Chain MonteâCarlo sampling"
8688:
if the reversed process is the same as the forward process.
4853:
Here is one method for doing so: first, define the function
3278:, ... to describe holding times in each of the states where
14177:
13814:
13241:
Pratas, D; Silva, R; Pinho, A; Ferreira, P (May 18, 2015).
12959:
11658:
11473:
11373:(1996). "Markov and the Birth of Chain Dependence Theory".
11336:
10784:
10351:
8869:
469:
15141:
Autoregressive conditional heteroskedasticity (ARCH) model
14075:
Carpinone, A; Giorgio, M; Langella, R.; Testa, A. (2015).
13463:
Champernowne, D (1953). "A model of income distribution".
12846:"Towards a Mathematical Theory of Cortical Micro-circuits"
12546:
11833:
11721:
11719:
11159:
10812:
Reuven Y. Rubinstein; Dirk P. Kroese (20 September 2011).
10805:
9778:
Markov chains are used throughout information processing.
8058:. Similarly we can construct such a dynamical system with
5066:(if exists) the stationary (or steady state) distribution
1819:{\displaystyle \Pr(X_{1}=x_{1},\ldots ,X_{n}=x_{n})>0.}
14406:. Appendix contains abridged Meyn & Tweedie. online:
13746:
Acemoglu, Daron; Georgy Egorov; Konstantin Sonin (2011).
11268:
11266:
11020:
Stochastic processes: a survey of the mathematical theory
9633:
times the probability a given molecule is in that state.
7524:{\displaystyle 1\to 2\to \dots \to n\to 1{\text{ and }}2}
4970:
Explain: The original matrix equation is equivalent to a
3228:
can be seen as measuring how quickly the transition from
14669:
Independent and identically distributed random variables
13868:
Formalized Music: Mathematics and Thought in Composition
13440:
The PageRank Citation Ranking: Bringing Order to the Web
12764:
12446:
9th European Signal Processing Conference (EUSIPCO 1998)
11243:
Hayes, Brian (2013). "First links in the Markov chain".
10204:
to be simulated and their parameters found numerically.
9280:) at intervals of ÎŽ units of time. The random variables
5016:
will have the 1 and the 0's in the same positions as in
4961:{\displaystyle \mathbf {Q} =f(\mathbf {0} _{n,n})^{-1}.}
3325:= 0, 1, 2, 3, ... and times indexed up to this value of
13555:
Fama, E (1965). "The behavior of stock market prices".
13291:
11857:
Donald L. Snyder; Michael I. Miller (6 December 2012).
11716:
11691:
11689:
11579:
Basic Principles and Applications of Probability Theory
5204:{\displaystyle \mathbf {P} =\mathbf {U\Sigma U} ^{-1}.}
4116:
Time-homogeneous Markov chain with a finite state space
2848:
There are three equivalent definitions of the process.
2235:
A Markov chain with memory (or a Markov chain of order
15146:
Autoregressive integrated moving average (ARIMA) model
14226:
Dynamic Probabilistic Systems, volume 1: Markov Chains
13517:
Bachelier, Louis (1900). "Théorie de la spéculation".
13176:
Mor, Bhavya; Garhwal, Sunita; Kumar, Ajay (May 2021).
13029:
11644:. Springer Science & Business Media. p. vii.
11582:. Springer Science & Business Media. p. 146.
11542:
11342:
11263:
11037:
10983:
10780:
10778:
9540:
7881:
6636:
if there is one communicating class, the state space.
4826:. Including the fact that the sum of each the rows in
4592:
4524:
4485:
4447:
4341:
4279:
4227:
4061:
4029:
3969:
1978:. The probability of the transition is independent of
1165:
of the various coin types on the table. For instance,
567:
Continuous-time Markov process or Markov jump process
410:, which are used for simulating sampling from complex
11863:. Springer Science & Business Media. p. 32.
11785:
11569:
11217:. Springer Science & Business Media. p. ix.
11206:
11204:
11202:
11175:
Charles Miller Grinstead; James Laurie Snell (1997).
10289:, the ratio of urban to rural residence, the rate of
10166:
10131:
10076:
10049:
10029:
10009:
9989:
9922:
9503:
9136:
9095:
9074:
9054:
9019:
8996:
8899:
8760:
8628:
8473:
8426:
8262:
8204:
8178:
8133:
8113:
8064:
7984:
7952:
7931:
7896:
7859:
7797:
7773:
7729:
7694:
7667:
7627:
7586:
7560:
7540:
7485:
7465:
7417:
7393:
7350:
7330:
7303:
7283:
7236:
7216:
7148:
7128:
7048:
7031:
if there are no outgoing transitions from the state.
6994:
6877:
6809:
6783:
6682:
6482:
5603:
5446:
5411:
5223:
5168:
5034:
4874:
4699:
4653:
4441:
4327:
4160:
4028:
3912:
3868:
3748:
3654:
3387:
3239:
3204:
3166:
3042:
2981:
2942:
2909:
2870:
2675:
2634:
2601:
2591:
In other words, the future state depends on the past
2249:
2211:
1991:
1860:
1745:
1546:
1445:
1400:
1354:
1327:
1300:
1255:
1216:
1171:
1140:
1109:
1079:
1052:
1025:
989:
962:
926:
872:
839:
808:
14525:
G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi,
13998:(November 1984). "A Travesty Generator for Micros".
12126:
11779:
11686:
11631:
10942:. Springer Science & Business Media. p. 7.
10931:
10929:
10160:
for all pages that are not linked to. The parameter
8353:{\displaystyle Y(t)={\big \{}X(s):s\in \,{\big \}}.}
8051:{\displaystyle T(X_{0},X_{1},\dots )=(X_{1},\dots )}
6611:(probabilistic cellular automata). See for instance
6594:
4570:(This example illustrates a periodic Markov chain.)
4128:-step transition probability can be computed as the
3316:
733:
derived in a 1928 paper an equation, now called the
13919:, IJCAI, pages 635â642, Barcelona, Spain, July 2011
13519:
Annales Scientifiques de l'Ăcole Normale SupĂ©rieure
12994:
12193:"Show that positive recurrence is a class property"
11726:Samuel Karlin; Howard E. Taylor (2 December 2012).
11637:
10990:Samuel Karlin; Howard E. Taylor (2 December 2012).
10865:
10775:
8460:that the chain enters one of the states in the set
1854:Time-homogeneous Markov chains are processes where
391:(CTMC). Markov processes are named in honor of the
14424:
14350:
14313:
14018:
13993:
13651:
13379:
12381:"An improvement of the Dulmage-Mendelsohn theorem"
12165:Stochastic Models in Operations Research, Volume 1
12048:
11447:, Seneta E, Crépel P, Fienberg SE, Gani J (eds.).
11293:
11199:
11064:
10866:
10172:
10152:
10117:
10062:
10035:
10015:
9995:
9963:
9888:and describe job arrivals, while transitions from
9690:use continuous-time Markov chains to describe the
9605:
9242:
9114:
9080:
9060:
9037:
9002:
8952:
8875:
8751:. These conditional probabilities may be found by
8669:
8597:
8444:
8352:
8241:be a non-Markovian process. Then define a process
8210:
8190:
8164:
8119:
8085:
8050:
7970:
7937:
7917:
7865:
7845:
7779:
7759:
7712:
7680:
7653:
7610:
7572:
7546:
7523:
7471:
7451:
7399:
7376:
7336:
7316:
7289:
7249:
7222:
7161:
7134:
7091:
7007:
6969:
6821:
6795:
6762:
6537:
6403:
5534:
5429:
5330:
5203:
5058:
4960:
4757:
4675:
4634:
4559:
4426:
4310:
4258:
4202:
4096:
4043:
3998:
3953:{\displaystyle \pi ={\frac {e}{\sum _{i}{e_{i}}}}}
3952:
3903:we see that the two concepts are related and that
3888:
3821:{\displaystyle p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).}
3820:
3692:
3617:
3220:
3182:
3152:
3016:
2967:
2928:
2883:
2761:
2653:
2620:
2583:
2224:
2189:
1966:
1818:
1727:
1482:
1431:
1385:
1340:
1313:
1286:
1240:
1202:
1153:
1122:
1092:
1065:
1038:
1011:
975:
948:
905:
858:
821:
600:While the time parameter is usually discrete, the
14377:(NB. This was originally published in Russian as
13906:"Finite-Length Markov Processes with Constraints"
13719:"A Markov Chain Example in Credit Risk Modelling"
13297:
11575:
11152:Meyn, S. Sean P., and Richard L. Tweedie. (2009)
11010:
10956:
10926:
9548:
9547:
9530:
9529:
9295:
4311:{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}
4259:{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}
2232:is a stationary distribution of the Markov chain.
1073:are impacted by our knowledge of values prior to
15886:
15028:Stochastic chains with memory of variable length
13182:Archives of Computational Methods in Engineering
12596:Gattringer, Christof; Lang, Christian B (2010).
12501:Ergodic Theory: Basic Examples and Constructions
10839:Dani Gamerman; Hedibert F. Lopes (10 May 2006).
10832:
6707:
6689:
5024:Convergence speed to the stationary distribution
5000:on its main diagonal that is equal to 1 and the
4602:
4281:
4229:
4162:
3765:
3388:
3043:
2800:is defined by a finite or countable state space
2411:
2260:
2085:
1992:
1914:
1861:
1746:
1665:
1547:
1439:state depends exclusively on the outcome of the
701:in 1907, and a branching process, introduced by
414:, and have found application in areas including
14155:Cambridge: National Bureau of Economic Research
13752:Proceedings of the National Academy of Sciences
13691:"Stock Price Volatility and the Equity Premium"
13175:
13140:
13070:
12903:Gupta, Ankur; Rawlings, James B. (April 2014).
11978:
11746:
11702:. John Wiley & Sons. pp. 373 and 374.
11272:
11156:. Cambridge University Press. (Preface, p. iii)
11043:
10261:Markov chains are generally used in describing
10070:links to it then it has transition probability
9817:. Markov chains also play an important role in
9742:
3835:sums to one and all elements are non-negative,
2773:
665:who claimed independence was necessary for the
14510:R. A. Sahner, K. S. Trivedi and A. Puliafito,
14492:, John Wiley & Sons, Inc. New York, 2002.
14259:Society for Industrial and Applied Mathematics
13385:
12595:
12505:Encyclopedia of Complexity and Systems Science
11839:
11812:
11695:
11596:
11511:
11509:
11210:
10962:
10935:
10859:
10785:Sean Meyn; Richard L. Tweedie (2 April 2009).
10610:limitation, a new approach has been proposed.
9828:for error correction), speech recognition and
9788:, which in a single step created the field of
9068:being a row vector, such that all elements in
8086:{\displaystyle \Omega =\Sigma ^{\mathbb {Z} }}
7918:{\displaystyle \Omega =\Sigma ^{\mathbb {N} }}
5118:be the diagonal matrix of left eigenvalues of
1509:A discrete-time Markov chain is a sequence of
14617:
14378:
13931:"MARKOV CHAIN MODELS: THEORETICAL BACKGROUND"
13619:Calvet, Laurent E.; Fisher, Adlai J. (2001).
13497:
12902:
12843:
12630:
12509:https://doi.org/10.1007/978-0-387-30440-3_177
11443:Bru B, Hertz S (2001). "Maurice Fréchet". In
11436:
11299:
11238:
11236:
11234:
11016:
8622:, the time-reversed process is defined to be
8342:
8280:
7723:(Dulmage-Mendelsohn theorem) The exponent is
5214:Let the eigenvalues be enumerated such that:
3899:By comparing this definition with that of an
3350:, ... and all states recorded at these times
1985:Stationary Markov chains are processes where
1498:
332:
14021:Virtual Muse: Experiments in Computer Poetry
13841:
13618:
13462:
13217:Studies in History and Philosophy of Science
12767:Journal of Chemical Information and Modeling
12732:Journal of Chemical Information and Modeling
12633:Design and Analysis of Biomolecular Circuits
12462:(1997). "Continuous-time Markov chains II".
12163:Heyman, Daniel P.; Sobel, Mathew J. (1982).
11819:. Courier Dover Publications. p. 7, 8.
11451:. New York, NY: Springer. pp. 331â334.
10658:
10270:
10023:is the number of known webpages, and a page
9350:Partially observable Markov decision process
9103:
9096:
8232:
6757:
6692:
2595:states. It is possible to construct a chain
900:
873:
12162:
11887:(1997). "Continuous-time Markov chains I".
11806:
11668:Bulletin of the London Mathematical Society
11506:
11488:Bulletin of the London Mathematical Society
11401:
11363:
10969:. Courier Dover Publications. p. 188.
9726:for disease outbreak and epidemic modeling.
4676:{\displaystyle \mathbf {QP} =\mathbf {Q} .}
2851:
1134:of the coins on the table, we could define
906:{\displaystyle \{X_{n}:n\in \mathbb {N} \}}
574:with the Markov property (for example, the
15156:Autoregressiveâmoving-average (ARMA) model
14624:
14610:
14025:. Hanover, NH: Wesleyan University Press.
13386:Langville, Amy N.; Meyer, Carl D. (2006).
13352:
12522:"Thermodynamics and Statistical Mechanics"
12411:
12248:. San Francisco: Holden-Day. p. 145.
11231:
11134:The Oxford Dictionary of Statistical Terms
11113:The Oxford Dictionary of Statistical Terms
10627:and differences when playing on grass vs.
9405:When the Markov matrix is replaced by the
9394:
8464:) is the minimal non-negative solution to
4690:from both sides and factoring then yields
3723:, called the transition matrix, with the (
339:
325:
16:Random process independent of past history
14241:(4). Translated by Link, David: 591â600.
14092:
13987:
13904:Pachet, F.; Roy, P.; Barbieri, G. (2011)
13791:
13781:
13771:
13665:
13587:
13538:
13516:
13447:
13414:
13274:
13055:
12936:
12879:
12869:
12706:
12680:
12572:
12499:Matthew Nicol and Karl Petersen, (2009) "
12436:
12396:
12111:
12003:
11988:
11753:Weiss, George H. (2006). "Random Walks".
11602:
11533:
11309:
10791:. Cambridge University Press. p. 3.
10634:
9843:to achieve very high compression ratios.
8403:
8339:
8077:
7909:
6575:Markov chains on a measurable state space
5525:
5414:
3017:{\displaystyle \left(X_{s}:s<t\right)}
896:
520:
14631:
14221:, 2-ya seriya, tom 15, pp. 135â156.
14139:
14110:"Quantitative Terrorism Risk Assessment"
14101:
14070:
14068:
14010:
13573:
13431:
13234:
12542:
12540:
12495:
12493:
12491:
11860:Random Point Processes in Time and Space
11442:
10900:Ăksendal, B. K. (Bernt Karsten) (2003).
10641:generate superficially real-looking text
10207:
9911:
9907:
9636:The classical model of enzyme activity,
9473:Markovian systems appear extensively in
8987:and setting all other elements to zero.
8695:
8249:represents a time-interval of states of
4022:The values of a stationary distribution
2855:
553:Markov chain on a measurable state space
468:
402:Markov chains have many applications as
18:
14473:Non-negative matrices and Markov chains
14415:Sequential Machines and Automata Theory
14396:Control Techniques for Complex Networks
14173:
14171:
14044:
14042:
14016:
13866:Xenakis, Iannis; Kanach, Sharon (1992)
13688:
13388:"A Reordering for the PageRank Problem"
13327:Control Techniques for Complex Networks
13309:MacTutor History of Mathematics Archive
13214:
12663:Du, Chao; Kou, S. C. (September 2012).
12519:
12452:
12271:
12091:
11922:
11918:
11916:
11877:
11638:William J. Anderson (6 December 2012).
11603:Bernstein, Jeremy (2005). "Bachelier".
9832:(such as in rearrangements detection).
9750:variability assessments are useful for
9523:
8253:. Mathematically, this takes the form:
7580:diagonal entries, then its exponent is
7277:, of a regular matrix, is the smallest
7264:
5610:
5083:is diagonalizable or equivalently that
5044:
5036:
4097:{\textstyle \sum _{i}1\cdot \pi _{i}=1}
4051:are associated with the state space of
2895:, and assume the process is in a state
796:
15887:
15462:Doob's martingale convergence theorems
14349:
14308:
14291:Markov Chains and Stochastic Stability
14232:
12458:
12351:
12299:
12243:
12215:
12167:. New York: McGraw-Hill. p. 230.
12156:
12130:; Kryukov, V. I.; Toom, A. L. (1978).
11925:Basics of Applied Stochastic Processes
11883:
11729:A First Course in Stochastic Processes
11515:
11407:
11369:
11181:. American Mathematical Soc. pp.
11154:Markov chains and stochastic stability
11072:The Cambridge Dictionary of Statistics
10996:. Academic Press. pp. 29 and 30.
10993:A First Course in Stochastic Processes
10818:. John Wiley & Sons. p. 225.
10788:Markov Chains and Stochastic Stability
10153:{\displaystyle {\frac {1-\alpha }{N}}}
9792:, opens by introducing the concept of
9785:A Mathematical Theory of Communication
9651:Also, the growth (and composition) of
8670:{\displaystyle {\hat {X}}_{t}=X_{T-t}}
6661:of the number of transitions by which
6568:
6386:
6309:
6238:
6167:
6068:
6015:
5968:
5879:
5841:
5809:
5455:
3889:{\displaystyle \pi \mathbf {P} =\pi .}
15214:Constant elasticity of variance (CEV)
15204:ChanâKarolyiâLongstaffâSanders (CKLS)
14605:
14591:A visual explanation of Markov Chains
14412:
14065:
13835:
13748:"Political model of social evolution"
13621:"Forecasting Multifractal Volatility"
12662:
12599:Quantum Chromodynamics on the Lattice
12537:
12488:
12358:. Internet Archive. New York, Wiley.
12306:. Internet Archive. New York, Wiley.
12267:
12265:
12006:"Convergence Rates for Markov Chains"
11752:
11576:Valeriy Skorokhod (5 December 2005).
11242:
11023:. Springer-Verlag. pp. 106â121.
10815:Simulation and the Monte Carlo Method
10639:Markov processes can also be used to
10125:for all pages that are linked to and
9773:
9758:
9323:System state is partially observable
9265:is found, it must be normalized to a
5028:As stated earlier, from the equation
3710:
3257:th jump of the process and variables
1845:called the state space of the chain.
1241:{\displaystyle 6\times 6\times 6=216}
14514:, Kluwer Academic Publishers, 1996.
14398:. Cambridge University Press, 2007.
14289:S. P. Meyn and R. L. Tweedie (1993)
14168:
14039:
13554:
13395:SIAM Journal on Scientific Computing
13319:
13105:
12635:, Springer New York, pp. 3â42,
12378:
11913:
11755:Encyclopedia of Statistical Sciences
11699:Probability and Stochastic Processes
10887:participating institution membership
7971:{\displaystyle T:\Omega \to \Omega }
6476:exponentially. This follows because
4822:is a stochastic matrix to solve for
4132:-th power of the transition matrix,
4124:is the same after each step, so the
3244:Define a discrete-time Markov chain
1493:
741:, starting in 1930s, and then later
639:conditional probability distribution
14527:Queueing Networks and Markov Chains
14107:
13336:, Cambridge University Press, 2007.
9357:
9038:{\displaystyle \varphi S=\varphi ,}
8702:stationary probability distribution
7888:measure-preserving dynamical system
7882:Measure-preserving dynamical system
7846:{\displaystyle \leq (d+1)+s(d+1-2)}
7257:are positive. Some authors call it
6860:) otherwise. For a recurrent state
4044:{\displaystyle \textstyle \pi _{i}}
3703:with initial condition P(0) is the
2243:is finite, is a process satisfying
591:continuous-time Markov chain (CTMC)
13:
15701:Skorokhod's representation theorem
15482:Law of large numbers (weak/strong)
13954:
13928:
12262:
11696:Ionut Florescu (7 November 2014).
10904:(6th ed.). Berlin: Springer.
10358:), or any other desirable metric.
10256:
10200:methods, allowing a wide range of
9846:
9586:
9563:
9413:, the resulting shift is termed a
8205:
8185:
8072:
8065:
7965:
7959:
7932:
7904:
7897:
7654:{\displaystyle \mathrm {sign} (M)}
7638:
7635:
7632:
7629:
7377:{\displaystyle \mathrm {sign} (M)}
7361:
7358:
7355:
7352:
6929:
6087:
4612:
4291:
4239:
4172:
3240:Jump chain/holding time definition
2975:is independent of previous values
1003:
940:
541:Continuous or general state space
14:
15921:
15671:Martingale representation theorem
14542:
14529:, John Wiley, 2nd edition, 2006.
13654:Journal of Financial Econometrics
12553:Psychonomic Bulletin & Review
12414:Foundations of modern probability
12187:
12094:"Interaction of Markov Processes"
10893:
10692:Markov chain approximation method
9900: > 1) occur at rate
9320:System state is fully observable
7034:
6601:locally interacting Markov chains
6595:Locally interacting Markov chains
6462:â â with a speed in the order of
5430:{\displaystyle \mathbb {R} ^{n},}
4006:) multiple of a left eigenvector
3643:first-order differential equation
3317:Transition probability definition
1483:{\displaystyle X_{n-1}=\ell ,m,p}
1012:{\displaystyle X_{7}\geq \$ 0.60}
587:discrete-time Markov chain (DTMC)
15716:Stochastic differential equation
15606:Doob's optional stopping theorem
15601:DoobâMeyer decomposition theorem
14278:. New York: John Wiley and Sons
13969:
13948:
13922:
13689:Brennan, Michael; Xiab, Yihong.
12803:Acta Crystallographica Section A
12669:The Annals of Applied Statistics
12437:Doblinger, G. (September 1998).
11792:. North-Holland. pp. 8â10.
11410:International Statistical Review
11375:International Statistical Review
11357:10.1111/j.1751-5823.2012.00181.x
11345:International Statistical Review
9115:{\displaystyle \|\varphi \|_{1}}
8608:
8420:of hitting times (where element
8384:
7452:{\displaystyle \leq (n-1)^{2}+1}
6579:
6375:
6298:
6227:
6156:
6057:
6004:
5957:
5907:
5895:
5892:
5868:
5830:
5798:
5758:
5746:
5743:
5740:
5711:
5708:
5705:
5679:
5676:
5673:
5650:
5647:
5644:
5634:
5497:
5449:
5185:
5182:
5179:
5170:
5049:
4929:
4920:
4891:
4876:
4736:
4718:
4709:
4701:
4666:
4658:
4655:
4619:
4594:
4420:
4397:
4375:
4329:
4298:
4246:
4193:
4179:
3999:{\textstyle \sum _{i}\pi _{i}=1}
3873:
2784:A continuous-time Markov chain (
367:of possible events in which the
45:
15586:Convergence of random variables
15472:FisherâTippettâGnedenko theorem
14505:SHARPE at the age of twenty-two
14231:Classical Text in Translation:
14081:Electric Power Systems Research
13898:
13880:
13860:
13808:
13739:
13711:
13682:
13645:
13612:
13567:
13547:
13510:
13491:
13456:
13339:
13208:
13169:
13134:
13099:
13064:
13023:
12988:
12953:
12896:
12837:
12793:
12758:
12723:
12656:
12624:
12589:
12513:
12430:
12405:
12372:
12345:
12320:
12293:
12237:
12209:
12181:
12136:. Manchester University Press.
12120:
12085:
12042:
11997:
11972:
11949:
11850:
11813:Emanuel Parzen (17 June 2015).
11763:10.1002/0471667196.ess2180.pub2
11279:. Wiley. pp. 235 and 358.
11211:Pierre Bremaud (9 March 2013).
11146:
11126:
11105:
11084:
11050:. Wiley. pp. 174 and 231.
10963:Emanuel Parzen (17 June 2015).
10682:Dynamics of Markovian particles
9694:present at a given site in the
9460:
9300:
7890:: Let the probability space be
6613:Interaction of Markov Processes
5510:
4983:One thing to notice is that if
4394:
4372:
1294:. The probability of achieving
609:
15184:Binomial options pricing model
14503:K. S. Trivedi and R.A.Sahner,
14427:Finite Mathematical Structures
12004:Rosenthal, Jeffrey S. (1995).
11786:Michael F. Shlesinger (1985).
11732:. Academic Press. p. 49.
11555:. Springer London. p. 5.
11449:Statisticians of the Centuries
11302:A Festschrift for Herman Rubin
10939:Applied Probability and Queues
10936:SĂžren Asmussen (15 May 2003).
10325:Markov chains are employed in
9579:
9550:
9525:
9425:on the subshift. Many chaotic
9296:Special types of Markov chains
9225:
9212:
9208:
9202:
9193:
9186:
9171:
9167:
9161:
9152:
8930:
8924:
8712:continuous-time Markov chain,
8636:
8336:
8333:
8327:
8318:
8312:
8306:
8294:
8288:
8272:
8266:
8153:
8147:
8045:
8026:
8020:
7988:
7962:
7840:
7822:
7813:
7801:
7754:
7742:
7648:
7642:
7507:
7501:
7495:
7489:
7434:
7421:
7407:be the number of states, then
7371:
7365:
7210:iff there exists some integer
7190:
7086:
7073:
6959:
6953:
6907:
6894:
6748:
6710:
6667:can be reached, starting from
6528:
6513:
6499:
6484:
5621:
5615:
5321:
5306:
5292:
5277:
5269:
5254:
5246:
5231:
4972:system of nĂn linear equations
4943:
4939:
4916:
4910:
4907:
4886:
4728:
4705:
4609:
4288:
4236:
4169:
3812:
3768:
3684:
3678:
3669:
3663:
3612:
3580:
3541:
3391:
3144:
3138:
3094:
3085:
3079:
3064:
3052:
3046:
2648:
2635:
2615:
2602:
2560:
2414:
2397:
2263:
2184:
2088:
2079:
1995:
1961:
1917:
1908:
1864:
1807:
1749:
1739:are well defined, that is, if
1719:
1668:
1659:
1550:
615:
112:Collectively exhaustive events
1:
15651:Kolmogorov continuity theorem
15487:Law of the iterated logarithm
14581:Introduction to Markov Chains
14433:Classical text. cf Chapter 6
14210:
14200:10.1016/j.solener.2019.04.014
14146:Woo, Gordon (December 2003).
13977:"Poet's Corner â Fieralingue"
13639:10.1016/S0304-4076(01)00069-0
13365:. IGI Global. pp. 448â.
13163:10.1016/j.solener.2018.05.055
13128:10.1016/S0038-092X(98)00004-8
13093:10.1016/j.solener.2018.07.056
13057:10.1016/j.solener.2015.02.032
13017:10.1016/j.solener.2014.02.026
12328:"10.3: Regular Markov Chains"
12221:"Markov Chains: Basic Theory"
10752:Stochastic cellular automaton
10327:algorithmic music composition
10234:Markov switching multifractal
10187:
9880: + 1 occur at rate
9389:stationary stochastic process
7760:{\displaystyle \leq n+s(n-2)}
7102:
6618:
5072:is a left eigenvector of row
4269:For some stochastic matrices
4143:. Additionally, in this case
1848:
949:{\displaystyle X_{6}=\$ 0.50}
572:continuous stochastic process
493:(sometimes characterized as "
480:
464:
15656:Kolmogorov extension theorem
15335:Generalized queueing network
14843:Interacting particle systems
14357:. Vol. II (122). 1965.
14148:"Insuring Against Al-Quaeda"
14145:
13957:"BASEBALL AS A MARKOV CHAIN"
12982:10.1016/0038-092X(88)90049-7
12871:10.1371/journal.pcbi.1000532
12472:10.1017/CBO9780511810633.005
12398:10.1016/0012-365X(95)00060-A
12300:Seneta, E. (Eugene) (1973).
12113:10.1016/0001-8708(70)90034-4
11957:"Chapter 11 "Markov Chains""
11897:10.1017/CBO9780511810633.004
11457:10.1007/978-1-4613-0179-0_71
9809:. They also allow effective
9768:automatic speech recognition
9743:Solar irradiance variability
9491:
9382:Ornstein isomorphism theorem
8743:of transitioning from state
8191:{\displaystyle S=\emptyset }
7092:{\displaystyle \pi _{i}=1/E}
6609:stochastic cellular automata
6427:(normalized by L2 norm) and
3183:{\displaystyle \delta _{ij}}
2780:Continuous-time Markov chain
2774:Continuous-time Markov chain
389:continuous-time Markov chain
7:
14788:Continuous-time random walk
14556:Encyclopedia of Mathematics
14114:The Journal of Risk Finance
13844:The Computer Music Tutorial
13229:10.1016/j.shpsa.2008.12.011
12641:10.1007/978-1-4419-6766-4_1
11605:American Journal of Physics
11178:Introduction to Probability
10762:Variable-order Markov model
10674:
10613:
10226:efficient-market hypothesis
10180:is taken to be about 0.15.
8718:embedded Markov chain (EMC)
8381:of order greater than one.
8165:{\displaystyle T^{-1}(S)=S}
7611:{\displaystyle \leq 2n-k-1}
7206:Some authors call a matrix
6605:interacting particle system
5378:is the left eigenvector of
3859:on it and so is defined by
3693:{\displaystyle P'(t)=P(t)Q}
1432:{\displaystyle X_{n}=i,j,k}
1386:{\displaystyle X_{2}=1,0,1}
1287:{\displaystyle X_{1}=0,1,0}
1203:{\displaystyle X_{6}=1,0,5}
748:
735:ChapmanâKolmogorov equation
514:discrete or continuous time
10:
15926:
15796:Extreme value theory (EVT)
15596:Doob decomposition theorem
14888:OrnsteinâUhlenbeck process
14659:Chinese restaurant process
14451:, D. van Nostrand Company
14293:. London: Springer-Verlag
14108:Woo, Gordon (2002-04-01).
14094:10.1016/j.epsr.2014.12.025
13842:Curtis Roads, ed. (1996).
13194:10.1007/s11831-020-09422-4
12197:Mathematics Stack Exchange
12071:10.1103/PhysRevE.84.041112
11123:(entry for "Markov chain")
10697:Markov chain geostatistics
9850:
9730:
9670:
9484:Markov chains are used in
9468:
9398:
9361:
9304:
8716:, is by first finding its
8700:One method of finding the
8388:
8245:, such that each state of
7384:as its adjacency matrix.
6572:
4834:equations for determining
3851:A stationary distribution
2777:
1505:Discrete-time Markov chain
1502:
1499:Discrete-time Markov chain
763:based on integers and the
752:
648:
381:discrete-time Markov chain
15864:
15768:
15676:Optional stopping theorem
15573:
15535:
15477:Large deviation principle
15444:
15358:
15315:
15282:
15229:HeathâJarrowâMorton (HJM)
15174:
15166:Moving-average (MA) model
15151:Autoregressive (AR) model
15131:
15041:
14976:Hidden Markov model (HMM)
14958:
14910:SchrammâLoewner evolution
14714:
14639:
14413:Booth, Taylor L. (1967).
14379:
14363:10.1007/978-3-662-25360-1
14334:10.1007/978-3-662-00031-1
14310:Dynkin, Eugene Borisovich
14247:10.1017/s0269889706001074
14017:Hartman, Charles (1996).
12823:10.1107/S0108767311044874
12608:10.1007/978-3-642-01850-3
12565:10.3758/s13423-016-1015-8
12412:Kallenberg, Olav (2002).
12379:Shen, Jian (1996-10-15).
11933:10.1007/978-3-540-89332-5
11923:Serfozo, Richard (2009).
11522:The Annals of Probability
10874:Oxford English Dictionary
10717:Markov information source
10707:Markov chain tree theorem
10659:Probabilistic forecasting
9638:MichaelisâMenten kinetics
9616:Michaelis-Menten kinetics
9439:ProuhetâThueâMorse system
9257:may be periodic, even if
8445:{\displaystyle k_{i}^{A}}
8233:Markovian representations
7713:{\displaystyle \leq 2n-2}
7297:such that all entries of
7230:such that all entries of
7142:such that all entries of
6433:is a probability vector,
5095:, one may start with the
4221:. If, by whatever means,
4010:of the transition matrix
2968:{\displaystyle X_{t+h}=j}
1737:conditional probabilities
1348:; for example, the state
755:Examples of Markov chains
745:, starting in the 1950s.
689:studied Markov chains on
667:weak law of large numbers
563:
545:
540:
537:
535:
412:probability distributions
15591:Doléans-Dade exponential
15421:Progressively measurable
15219:CoxâIngersollâRoss (CIR)
13758:(Suppl 4): 21292â21296.
13314:University of St Andrews
12244:Parzen, Emanuel (1962).
11846:. Wiley. p. 46, 47.
11273:Sheldon M. Ross (1996).
11044:Sheldon M. Ross (1996).
10768:
10757:Telescoping Markov chain
10702:Markov chain mixing time
10354:note values, frequency (
10320:
10304:
10194:Markov chain Monte Carlo
9979:of a webpage as used by
9706:matrix population models
9415:topological Markov chain
9081:{\displaystyle \varphi }
9061:{\displaystyle \varphi }
9003:{\displaystyle \varphi }
8979:formed by selecting the
7791:. It can be improved to
4219:PerronâFrobenius theorem
3289:exponential distribution
2852:Infinitesimal definition
408:Markov chain Monte Carlo
282:Law of total probability
277:Conditional independence
166:Exponential distribution
151:Probability distribution
15811:Mathematical statistics
15801:Large deviations theory
15631:Infinitesimal generator
15492:Maximal ergodic theorem
15411:Piecewise-deterministic
15013:Random dynamical system
14878:Markov additive process
14006:(12): 129â131, 449â469.
13829:10.1162/014892699559733
13783:10.1073/pnas.1019454108
13626:Journal of Econometrics
13540:2027/coo.31924001082803
12099:Advances in Mathematics
12092:Spitzer, Frank (1970).
11843:Stochastipoic processes
11840:Joseph L. Doob (1990).
11516:Cramér, Harald (1976).
11320:10.1214/lnms/1196285381
10879:Oxford University Press
10712:Markov decision process
10202:posterior distributions
10173:{\displaystyle \alpha }
9688:models of DNA evolution
9419:subshift of finite type
9401:Subshift of finite type
9395:Subshift of finite type
9345:Markov decision process
9088:are greater than 0 and
8741:conditional probability
8408:For a subset of states
8391:Phase-type distribution
8211:{\displaystyle \Omega }
7978:be the shift operator:
7938:{\displaystyle \Sigma }
7573:{\displaystyle k\geq 1}
7199:, defined according to
6659:greatest common divisor
6623:Two states are said to
4861:) to return the matrix
4644:It is always true that
3841:right stochastic matrix
3637:is the solution of the
2929:{\displaystyle X_{t}=i}
2654:{\displaystyle (X_{n})}
2621:{\displaystyle (Y_{n})}
1829:The possible values of
859:{\displaystyle X_{0}=0}
261:Conditional probability
15910:Random text generation
15646:KarhunenâLoĂšve theorem
15581:CameronâMartin formula
15545:BurkholderâDavisâGundy
14940:Variance gamma process
14228:. John Wiley and Sons.
13817:Computer Music Journal
13676:10.1093/jjfinec/nbh003
12520:Fitzpatrick, Richard.
12332:Mathematics LibreTexts
11535:10.1214/aop/1176996025
11017:John Lamperti (1977).
10635:Markov text generators
10271:
10251:Credit rating agencies
10174:
10154:
10119:
10064:
10037:
10017:
9997:
9972:
9965:
9841:Lempel-Ziv compression
9819:reinforcement learning
9607:
9380:Note, however, by the
9244:
9116:
9082:
9062:
9039:
9004:
8954:
8877:
8690:Kolmogorov's criterion
8684:A chain is said to be
8671:
8599:
8446:
8404:Expected hitting times
8354:
8212:
8192:
8166:
8121:
8087:
8052:
7972:
7939:
7919:
7867:
7847:
7781:
7761:
7714:
7682:
7655:
7612:
7574:
7548:
7525:
7473:
7453:
7401:
7378:
7338:
7318:
7291:
7251:
7224:
7163:
7136:
7093:
7009:
6971:
6933:
6823:
6797:
6796:{\displaystyle k>1}
6764:
6539:
6405:
5536:
5484:
5431:
5332:
5205:
5060:
4962:
4759:
4677:
4636:
4561:
4428:
4312:
4260:
4204:
4098:
4045:
4000:
3954:
3890:
3822:
3715:If the state space is
3694:
3619:
3222:
3221:{\displaystyle q_{ij}}
3184:
3154:
3018:
2969:
2930:
2885:
2861:
2806:transition rate matrix
2763:
2655:
2622:
2585:
2226:
2191:
1968:
1820:
1729:
1484:
1433:
1387:
1342:
1315:
1288:
1242:
1204:
1155:
1124:
1094:
1067:
1040:
1013:
977:
956:. If we know not just
950:
907:
860:
823:
538:Countable state space
521:Types of Markov chains
485:A Markov process is a
477:
473:Russian mathematician
203:Continuous or discrete
156:Bernoulli distribution
24:
15776:Actuarial mathematics
15738:Uniform integrability
15733:Stratonovich integral
15661:LĂ©vyâProkhorov metric
15565:MarcinkiewiczâZygmund
15452:Central limit theorem
15054:Gaussian random field
14883:McKeanâVlasov process
14803:Dyson Brownian motion
14664:GaltonâWatson process
14385:Markovskiye protsessy
13347:U.S. patent 6,285,999
11070:Everitt, B.S. (2002)
10208:Economics and finance
10175:
10155:
10120:
10065:
10063:{\displaystyle k_{i}}
10038:
10018:
9998:
9966:
9915:
9908:Internet applications
9782:'s famous 1948 paper
9608:
9479:statistical mechanics
9341:System is controlled
9328:System is autonomous
9245:
9117:
9083:
9063:
9040:
9005:
8955:
8878:
8739:, and represents the
8696:Embedded Markov chain
8672:
8600:
8447:
8355:
8213:
8193:
8167:
8122:
8088:
8053:
7973:
7940:
7920:
7875:diameter of the graph
7868:
7848:
7782:
7762:
7715:
7683:
7681:{\displaystyle M^{2}}
7656:
7613:
7575:
7549:
7526:
7474:
7454:
7402:
7379:
7339:
7319:
7317:{\displaystyle M^{k}}
7292:
7252:
7250:{\displaystyle M^{k}}
7225:
7164:
7162:{\displaystyle M^{k}}
7137:
7094:
7010:
7008:{\displaystyle M_{i}}
6972:
6913:
6824:
6798:
6765:
6540:
6406:
5537:
5464:
5432:
5333:
5206:
5079:. Then assuming that
5061:
4963:
4760:
4678:
4637:
4562:
4429:
4313:
4261:
4205:
4099:
4046:
4001:
3955:
3891:
3823:
3695:
3620:
3291:with rate parameter â
3223:
3185:
3155:
3019:
2970:
2931:
2886:
2884:{\displaystyle X_{t}}
2859:
2764:
2656:
2623:
2586:
2227:
2225:{\displaystyle X_{0}}
2192:
1969:
1821:
1730:
1485:
1434:
1388:
1343:
1341:{\displaystyle X_{1}}
1316:
1314:{\displaystyle X_{2}}
1289:
1243:
1205:
1156:
1154:{\displaystyle X_{n}}
1125:
1123:{\displaystyle X_{n}}
1095:
1093:{\displaystyle X_{6}}
1068:
1066:{\displaystyle X_{7}}
1041:
1039:{\displaystyle X_{6}}
1014:
978:
976:{\displaystyle X_{6}}
951:
908:
861:
824:
822:{\displaystyle X_{n}}
712:Irénée-Jules Bienaymé
680:central limit theorem
472:
161:Binomial distribution
22:
15851:Time series analysis
15806:Mathematical finance
15691:Reflection principle
15018:Regenerative process
14818:FlemingâViot process
14633:Stochastic processes
14449:Finite Markov Chains
14435:Finite Markov Chains
14322:Majone, Giandomenico
14276:Stochastic Processes
14253:Leo Breiman (1992)
13983:on December 6, 2010.
13465:The Economic Journal
13442:(Technical report).
13300:Robertson, Edmund F.
12466:. pp. 108â127.
12385:Discrete Mathematics
12246:Stochastic Processes
11927:. Berlin: Springer.
11816:Stochastic Processes
11680:10.1112/blms/22.1.31
11500:10.1112/blms/22.1.31
11276:Stochastic processes
11092:Stochastic Processes
11047:Stochastic processes
10966:Stochastic Processes
10742:Quantum Markov chain
10687:GaussâMarkov process
10665:stochastic terrorism
10279:economic development
10164:
10129:
10074:
10047:
10027:
10007:
9987:
9920:
9896: â 1 (for
9764:Hidden Markov models
9724:Compartmental models
9501:
9455:block-coding systems
9451:context-free systems
9134:
9093:
9072:
9052:
9017:
8994:
8897:
8758:
8626:
8471:
8456:, starting in state
8424:
8260:
8218:(up to a null set).
8202:
8176:
8131:
8111:
8062:
7982:
7950:
7929:
7894:
7857:
7795:
7771:
7727:
7692:
7665:
7625:
7584:
7558:
7538:
7483:
7463:
7415:
7391:
7348:
7328:
7301:
7281:
7271:index of primitivity
7265:Index of primitivity
7234:
7214:
7146:
7126:
7046:
6992:
6875:
6807:
6781:
6680:
6480:
5601:
5444:
5409:
5221:
5166:
5032:
5012:th row or column of
4872:
4697:
4651:
4590:
4439:
4325:
4277:
4225:
4158:
4059:
4026:
3967:
3910:
3866:
3746:
3652:
3385:
3378:, ... it holds that
3202:
3164:
3040:
2979:
2940:
2907:
2868:
2673:
2632:
2599:
2247:
2209:
1989:
1858:
1743:
1544:
1443:
1398:
1352:
1325:
1298:
1253:
1214:
1169:
1138:
1107:
1077:
1050:
1023:
987:
960:
924:
870:
866:, then the sequence
837:
806:
797:A non-Markov example
772:, also known as the
714:. Starting in 1928,
707:Henry William Watson
387:process is called a
287:Law of large numbers
256:Marginal probability
181:Poisson distribution
30:Part of a series on
15846:Stochastic analysis
15686:Quadratic variation
15681:Prokhorov's theorem
15616:FeynmanâKac formula
15086:Markov random field
14734:Birthâdeath process
14380:ĐĐ°ŃĐșĐŸĐČŃĐșОД ĐżŃĐŸŃĐ”ŃŃŃ
14192:2019SoEn..184..688M
13870:, Pendragon Press.
13764:2011PNAS..10821292A
13726:Columbia University
13557:Journal of Business
13407:2006SJSC...27.2112L
13298:O'Connor, John J.;
13259:2015NatSR...510203P
13155:2018SoEn..170..174M
13120:1998SoEn...62..101M
13085:2018SoEn..173..487M
13048:2015SoEn..115..229B
13009:2014SoEn..103..160N
12974:1988SoEn...40..269A
12921:2014AIChE..60.1253G
12862:2009PLSCB...5E0532G
12815:2012AcCrA..68..148K
12691:2012arXiv1209.6210D
12063:2011PhRvE..84d1112F
11891:. pp. 60â107.
11617:2005AmJPh..73..395B
11257:10.1511/2013.101.92
10877:(Online ed.).
10747:Semi-Markov process
10732:Markov random field
10446:
10363:
10245:general equilibrium
9863:Agner Krarup Erlang
9815:pattern recognition
9805:techniques such as
9702:Population dynamics
9536:
9335:Hidden Markov model
8565:
8492:
8441:
7661:is symmetric, then
6963:
6822:{\displaystyle k=1}
6569:General state space
6391:
6314:
6243:
6172:
6138:
6073:
6054:
6020:
6001:
5973:
5954:
5884:
5846:
5814:
4586:matrix, and define
489:that satisfies the
416:Bayesian statistics
246:Complementary event
188:Probability measure
176:Pareto distribution
171:Normal distribution
15816:Probability theory
15696:Skorokhod integral
15666:Malliavin calculus
15249:Korn-Kreer-Lenssen
15133:Time series models
15096:PitmanâYor process
14573:2008-05-22 at the
14235:Science in Context
13911:2012-04-14 at the
13735:on March 24, 2016.
13531:10.24033/asens.476
13332:2015-05-13 at the
13325:S. P. Meyn, 2007.
13247:Scientific Reports
12699:10.1214/12-aoas541
11304:. pp. 75â91.
11245:American Scientist
11090:Parzen, E. (1962)
10444:
10361:
10347:probability vector
10329:, particularly in
10311:Snakes and Ladders
10214:D. G. Champernowne
10198:Bayesian inference
10170:
10150:
10115:
10060:
10033:
10013:
9993:
9973:
9961:
9790:information theory
9774:Information theory
9766:have been used in
9759:Speech recognition
9603:
9571:
9555:
9240:
9112:
9078:
9058:
9035:
9000:
8950:
8890:may be written as
8873:
8868:
8816:
8667:
8595:
8593:
8551:
8537:
8478:
8442:
8427:
8350:
8208:
8188:
8162:
8117:
8083:
8048:
7968:
7935:
7915:
7863:
7843:
7789:girth of the graph
7777:
7757:
7710:
7678:
7651:
7608:
7570:
7544:
7521:
7469:
7449:
7397:
7374:
7334:
7314:
7287:
7247:
7220:
7159:
7132:
7089:
7005:
6986:positive recurrent
6967:
6940:
6844:if, starting from
6819:
6793:
6760:
6535:
6401:
6399:
6373:
6296:
6225:
6154:
6124:
6055:
6040:
6002:
5987:
5955:
5940:
5866:
5828:
5796:
5559:. In other words,
5532:
5427:
5382:corresponding to λ
5328:
5201:
5158:eigendecomposition
5097:Jordan normal form
5093:defective matrices
5056:
4958:
4755:
4673:
4632:
4616:
4557:
4551:
4510:
4474:
4424:
4366:
4308:
4295:
4256:
4243:
4200:
4176:
4094:
4071:
4041:
4040:
3996:
3979:
3950:
3934:
3886:
3831:Since each row of
3818:
3711:Finite state space
3690:
3615:
3218:
3180:
3150:
3014:
2965:
2926:
2881:
2862:
2759:
2651:
2618:
2581:
2579:
2222:
2187:
1964:
1816:
1725:
1480:
1429:
1383:
1338:
1311:
1284:
1238:
1200:
1151:
1120:
1090:
1063:
1036:
1009:
973:
946:
917:a Markov process.
903:
856:
819:
606:countably infinite
487:stochastic process
478:
436:information theory
404:statistical models
377:countably infinite
361:stochastic process
297:Boole's inequality
233:Stochastic process
122:Mutual exclusivity
39:Probability theory
25:
15882:
15881:
15836:Signal processing
15555:Doob's upcrossing
15550:Doob's martingale
15514:EngelbertâSchmidt
15457:Donsker's theorem
15391:Feller-continuous
15259:RendlemanâBartter
15049:Dirichlet process
14966:Branching process
14935:Telegraph process
14828:Geometric process
14808:Empirical process
14798:Diffusion process
14654:Branching process
14649:Bernoulli process
14535:978-0-7923-9650-5
14486:Kishor S. Trivedi
14481:978-0-387-29765-1
14404:978-0-521-88441-9
14375:. Title-No. 5105.
14372:978-3-662-23320-7
14346:. Title-No. 5104.
14343:978-3-662-00033-5
14268:. (See Chapter 7)
14032:978-0-8195-2239-9
13894:on July 13, 2012.
13853:978-0-262-18158-7
13425:10.1137/040607551
13372:978-1-5225-0106-0
13267:10.1038/srep10203
12929:10.1002/aic.14409
12809:(Pt 1): 148â155.
12779:10.1021/ci9000458
12744:10.1021/ci9000458
12617:978-3-642-01849-7
12423:978-0-387-95313-7
12365:978-0-470-77605-6
12313:978-0-470-77605-6
12051:Physical Review E
11942:978-3-540-89331-8
11870:978-1-4612-3166-0
11826:978-0-486-79688-8
11799:978-0-444-86937-1
11739:978-0-08-057041-9
11709:978-1-118-59320-2
11651:978-1-4612-3038-0
11625:10.1119/1.1848117
11589:978-3-540-26312-8
11562:978-1-4471-7262-8
11466:978-0-387-95283-3
11329:978-0-940600-61-4
11286:978-0-471-12062-9
11224:978-1-4757-3124-8
11192:978-0-8218-0749-1
11111:Dodge, Y. (2003)
11057:978-0-471-12062-9
11030:978-3-540-90275-1
11003:978-0-08-057041-9
10976:978-0-486-79688-8
10949:978-0-387-00211-8
10885:(Subscription or
10852:978-1-58488-587-0
10825:978-1-118-21052-9
10798:978-0-521-73182-9
10651:, Jeff Harrison,
10649:dissociated press
10589:
10588:
10445:2nd-order matrix
10442:
10441:
10362:1st-order matrix
10299:democratic regime
10238:Laurent E. Calvet
10230:James D. Hamilton
10184:individual user.
10148:
10113:
10092:
10036:{\displaystyle i}
10016:{\displaystyle N}
9996:{\displaystyle i}
9959:
9938:
9826:Viterbi algorithm
9807:arithmetic coding
9601:
9594:
9593:
9592:
9589:
9584:
9578:
9570:
9569:
9566:
9561:
9557:
9518:
9514:
9508:
9427:dynamical systems
9375:Bernoulli process
9355:
9354:
9235:
8861:
8838:
8831:
8801:
8639:
8577:
8522:
8504:
8120:{\displaystyle S}
7866:{\displaystyle d}
7780:{\displaystyle s}
7547:{\displaystyle M}
7516:
7472:{\displaystyle M}
7400:{\displaystyle n}
7337:{\displaystyle M}
7290:{\displaystyle k}
7223:{\displaystyle k}
7197:ergodic processes
7135:{\displaystyle k}
6829:and the state is
6361:
6284:
6213:
6103:
5369:matrix, that is,
5074:stochastic matrix
4976:stochastic matrix
4816:stochastic matrix
4601:
4547:
4535:
4470:
4458:
4280:
4228:
4161:
4062:
3970:
3963:is a normalized (
3948:
3925:
3196:little-o notation
2903:. Then, knowing
2566:
1494:Formal definition
1161:to represent the
1130:to represent the
776:process, and the
722:Andrey Kolmogorov
699:Tatyana Ehrenfest
682:for such chains.
676:Alexander Pushkin
622:transition matrix
582:
581:
448:speech processing
444:signal processing
349:
348:
251:Joint probability
198:Bernoulli process
97:Probability space
15917:
15895:Markov processes
15856:Machine learning
15743:Usual hypotheses
15626:Girsanov theorem
15611:Dynkin's formula
15376:Continuous paths
15284:Actuarial models
15224:GarmanâKohlhagen
15194:BlackâKarasinski
15189:BlackâDermanâToy
15176:Financial models
15042:Fields and other
14971:Gaussian process
14920:Sigma-martingale
14724:Additive process
14626:
14619:
14612:
14603:
14602:
14582:
14564:
14432:
14430:
14418:
14382:
14381:
14376:
14356:
14353:Markov Processes
14347:
14319:
14316:Markov Processes
14250:
14204:
14203:
14175:
14166:
14165:
14163:
14161:
14152:
14143:
14137:
14136:
14134:
14132:
14126:10.1108/eb022949
14105:
14099:
14098:
14096:
14072:
14063:
14062:
14055:Energy Economics
14046:
14037:
14036:
14024:
14014:
14008:
14007:
13996:O'Rourke, Joseph
13991:
13985:
13984:
13979:. Archived from
13973:
13967:
13966:
13964:
13963:
13955:Pankin, Mark D.
13952:
13946:
13945:
13943:
13942:
13933:. Archived from
13929:Pankin, Mark D.
13926:
13920:
13902:
13896:
13895:
13890:. Archived from
13884:
13878:
13864:
13858:
13857:
13839:
13833:
13832:
13812:
13806:
13805:
13795:
13785:
13775:
13743:
13737:
13736:
13734:
13728:. Archived from
13723:
13715:
13709:
13708:
13706:
13700:. Archived from
13695:
13686:
13680:
13679:
13669:
13649:
13643:
13642:
13616:
13610:
13609:
13591:
13571:
13565:
13564:
13551:
13545:
13544:
13542:
13514:
13508:
13507:
13495:
13489:
13488:
13460:
13454:
13453:
13451:
13435:
13429:
13428:
13418:
13401:(6): 2112â2113.
13392:
13383:
13377:
13376:
13356:
13350:
13349:
13343:
13337:
13323:
13317:
13316:
13295:
13289:
13288:
13278:
13253:(10203): 10203.
13238:
13232:
13231:
13212:
13206:
13205:
13188:(3): 1429â1448.
13173:
13167:
13166:
13138:
13132:
13131:
13103:
13097:
13096:
13068:
13062:
13061:
13059:
13027:
13021:
13020:
12992:
12986:
12985:
12957:
12951:
12950:
12940:
12915:(4): 1253â1268.
12900:
12894:
12893:
12883:
12873:
12856:(10): e1000532.
12850:PLOS Comput Biol
12841:
12835:
12834:
12797:
12791:
12790:
12773:(7): 1630â1642.
12762:
12756:
12755:
12738:(7): 1630â1642.
12727:
12721:
12720:
12710:
12684:
12660:
12654:
12653:
12628:
12622:
12621:
12593:
12587:
12586:
12576:
12544:
12535:
12534:
12532:
12531:
12526:
12517:
12511:
12497:
12486:
12485:
12456:
12450:
12449:
12443:
12434:
12428:
12427:
12409:
12403:
12402:
12400:
12376:
12370:
12369:
12349:
12343:
12342:
12340:
12339:
12324:
12318:
12317:
12297:
12291:
12290:
12288:
12287:
12277:"Ergodic Theory"
12269:
12260:
12259:
12241:
12235:
12234:
12232:
12230:
12225:
12213:
12207:
12206:
12204:
12203:
12185:
12179:
12178:
12160:
12154:
12153:
12151:
12150:
12128:Dobrushin, R. L.
12124:
12118:
12117:
12115:
12089:
12083:
12082:
12046:
12040:
12039:
12037:
12036:
12001:
11995:
11994:
11992:
11976:
11970:
11969:
11967:
11966:
11961:
11953:
11947:
11946:
11920:
11911:
11910:
11881:
11875:
11874:
11854:
11848:
11847:
11837:
11831:
11830:
11810:
11804:
11803:
11783:
11777:
11776:
11750:
11744:
11743:
11723:
11714:
11713:
11693:
11684:
11683:
11662:
11656:
11655:
11635:
11629:
11628:
11600:
11594:
11593:
11573:
11567:
11566:
11546:
11540:
11539:
11537:
11513:
11504:
11503:
11482:
11471:
11470:
11440:
11434:
11433:
11405:
11399:
11398:
11367:
11361:
11360:
11340:
11334:
11333:
11313:
11297:
11291:
11290:
11270:
11261:
11260:
11240:
11229:
11228:
11208:
11197:
11196:
11172:
11157:
11150:
11144:
11130:
11124:
11109:
11103:
11088:
11082:
11068:
11062:
11061:
11041:
11035:
11034:
11014:
11008:
11007:
10987:
10981:
10980:
10960:
10954:
10953:
10933:
10924:
10923:
10897:
10891:
10890:
10882:
10870:
10863:
10857:
10856:
10836:
10830:
10829:
10809:
10803:
10802:
10782:
10669:solar irradiance
10647:" software (see
10645:parody generator
10447:
10443:
10429:
10428:
10410:
10409:
10386:
10385:
10378:
10377:
10364:
10360:
10276:
10218:Herbert A. Simon
10179:
10177:
10176:
10171:
10159:
10157:
10156:
10151:
10149:
10144:
10133:
10124:
10122:
10121:
10116:
10114:
10109:
10098:
10093:
10091:
10090:
10078:
10069:
10067:
10066:
10061:
10059:
10058:
10042:
10040:
10039:
10034:
10022:
10020:
10019:
10014:
10002:
10000:
9999:
9994:
9970:
9968:
9967:
9962:
9960:
9955:
9944:
9939:
9937:
9936:
9924:
9811:state estimation
9803:entropy encoding
9799:data compression
9748:Solar irradiance
9612:
9610:
9609:
9604:
9602:
9599:
9595:
9590:
9587:
9585:
9582:
9576:
9574:
9572:
9567:
9564:
9562:
9559:
9558:
9556:
9554:
9553:
9546:
9538:
9537:
9535:
9528:
9520:
9516:
9509:
9506:
9435:closed manifolds
9407:adjacency matrix
9370:Bernoulli scheme
9364:Bernoulli scheme
9358:Bernoulli scheme
9315:
9314:
9264:
9249:
9247:
9246:
9241:
9236:
9234:
9233:
9228:
9224:
9223:
9222:
9183:
9182:
9181:
9144:
9127:may be found as
9126:
9123:= 1. From this,
9121:
9119:
9118:
9113:
9111:
9110:
9087:
9085:
9084:
9079:
9067:
9065:
9064:
9059:
9044:
9042:
9041:
9036:
9009:
9007:
9006:
9001:
8983:from the matrix
8959:
8957:
8956:
8951:
8946:
8945:
8937:
8933:
8882:
8880:
8879:
8874:
8872:
8871:
8862:
8859:
8839:
8836:
8832:
8830:
8829:
8828:
8815:
8799:
8798:
8786:
8773:
8772:
8730:, is denoted by
8707:
8676:
8674:
8673:
8668:
8666:
8665:
8647:
8646:
8641:
8640:
8632:
8604:
8602:
8601:
8596:
8594:
8578:
8575:
8564:
8559:
8550:
8549:
8536:
8505:
8502:
8491:
8486:
8451:
8449:
8448:
8443:
8440:
8435:
8359:
8357:
8356:
8351:
8346:
8345:
8284:
8283:
8217:
8215:
8214:
8209:
8197:
8195:
8194:
8189:
8171:
8169:
8168:
8163:
8146:
8145:
8126:
8124:
8123:
8118:
8092:
8090:
8089:
8084:
8082:
8081:
8080:
8057:
8055:
8054:
8049:
8038:
8037:
8013:
8012:
8000:
7999:
7977:
7975:
7974:
7969:
7944:
7942:
7941:
7936:
7924:
7922:
7921:
7916:
7914:
7913:
7912:
7872:
7870:
7869:
7864:
7852:
7850:
7849:
7844:
7786:
7784:
7783:
7778:
7766:
7764:
7763:
7758:
7719:
7717:
7716:
7711:
7687:
7685:
7684:
7679:
7677:
7676:
7660:
7658:
7657:
7652:
7641:
7617:
7615:
7614:
7609:
7579:
7577:
7576:
7571:
7553:
7551:
7550:
7545:
7530:
7528:
7527:
7522:
7517:
7514:
7478:
7476:
7475:
7470:
7458:
7456:
7455:
7450:
7442:
7441:
7411:The exponent is
7406:
7404:
7403:
7398:
7383:
7381:
7380:
7375:
7364:
7343:
7341:
7340:
7335:
7323:
7321:
7320:
7315:
7313:
7312:
7296:
7294:
7293:
7288:
7256:
7254:
7253:
7248:
7246:
7245:
7229:
7227:
7226:
7221:
7184: = 1.
7168:
7166:
7165:
7160:
7158:
7157:
7141:
7139:
7138:
7133:
7098:
7096:
7095:
7090:
7085:
7084:
7069:
7058:
7057:
7014:
7012:
7011:
7006:
7004:
7003:
6976:
6974:
6973:
6968:
6962:
6951:
6932:
6927:
6906:
6905:
6887:
6886:
6828:
6826:
6825:
6820:
6802:
6800:
6799:
6794:
6769:
6767:
6766:
6761:
6741:
6740:
6722:
6721:
6672:
6666:
6656:
6650:
6644:
6563:
6544:
6542:
6541:
6536:
6531:
6526:
6525:
6516:
6502:
6497:
6496:
6487:
6456:
6437:
6431:
6418:
6410:
6408:
6407:
6402:
6400:
6396:
6392:
6390:
6389:
6383:
6378:
6372:
6371:
6366:
6362:
6360:
6359:
6350:
6349:
6340:
6333:
6332:
6313:
6312:
6306:
6301:
6295:
6294:
6289:
6285:
6283:
6282:
6273:
6272:
6263:
6256:
6255:
6242:
6241:
6235:
6230:
6224:
6223:
6218:
6214:
6212:
6211:
6202:
6201:
6192:
6185:
6184:
6171:
6170:
6164:
6159:
6153:
6152:
6137:
6132:
6117:
6104:
6101:
6099:
6098:
6086:
6085:
6075:
6072:
6071:
6065:
6060:
6053:
6048:
6039:
6038:
6019:
6018:
6012:
6007:
6000:
5995:
5986:
5985:
5972:
5971:
5965:
5960:
5953:
5948:
5939:
5938:
5923:
5919:
5918:
5910:
5904:
5903:
5898:
5889:
5885:
5883:
5882:
5876:
5871:
5865:
5864:
5845:
5844:
5838:
5833:
5827:
5826:
5813:
5812:
5806:
5801:
5795:
5794:
5774:
5770:
5769:
5761:
5755:
5754:
5749:
5731:
5727:
5723:
5722:
5714:
5695:
5691:
5690:
5682:
5666:
5662:
5661:
5653:
5637:
5625:
5624:
5613:
5594:â â. That means
5563:
5557:
5541:
5539:
5538:
5533:
5528:
5520:
5519:
5506:
5505:
5500:
5494:
5493:
5483:
5478:
5460:
5459:
5458:
5452:
5436:
5434:
5433:
5428:
5423:
5422:
5417:
5350:
5337:
5335:
5334:
5329:
5324:
5319:
5318:
5309:
5295:
5290:
5289:
5280:
5272:
5267:
5266:
5257:
5249:
5244:
5243:
5234:
5210:
5208:
5207:
5202:
5197:
5196:
5188:
5173:
5070:
5065:
5063:
5062:
5057:
5052:
5047:
5039:
4967:
4965:
4964:
4959:
4954:
4953:
4938:
4937:
4932:
4923:
4906:
4905:
4894:
4879:
4830:is 1, there are
4764:
4762:
4761:
4756:
4751:
4750:
4739:
4727:
4726:
4721:
4712:
4704:
4682:
4680:
4679:
4674:
4669:
4661:
4641:
4639:
4638:
4633:
4628:
4627:
4622:
4615:
4597:
4566:
4564:
4563:
4558:
4556:
4555:
4548:
4540:
4536:
4528:
4515:
4514:
4479:
4478:
4471:
4463:
4459:
4451:
4433:
4431:
4430:
4425:
4423:
4415:
4414:
4400:
4387:
4386:
4378:
4371:
4370:
4332:
4317:
4315:
4314:
4309:
4307:
4306:
4301:
4294:
4265:
4263:
4262:
4257:
4255:
4254:
4249:
4242:
4209:
4207:
4206:
4201:
4196:
4188:
4187:
4182:
4175:
4150:
4142:
4104:we see that the
4103:
4101:
4100:
4095:
4087:
4086:
4070:
4050:
4048:
4047:
4042:
4039:
4038:
4005:
4003:
4002:
3997:
3989:
3988:
3978:
3959:
3957:
3956:
3951:
3949:
3947:
3946:
3945:
3944:
3933:
3920:
3895:
3893:
3892:
3887:
3876:
3854:
3827:
3825:
3824:
3819:
3805:
3804:
3786:
3785:
3761:
3760:
3699:
3697:
3696:
3691:
3662:
3639:forward equation
3624:
3622:
3621:
3616:
3611:
3610:
3598:
3597:
3579:
3578:
3577:
3576:
3561:
3560:
3540:
3539:
3527:
3526:
3525:
3524:
3501:
3500:
3488:
3487:
3486:
3485:
3468:
3467:
3455:
3454:
3453:
3452:
3435:
3434:
3416:
3415:
3414:
3413:
3253:to describe the
3227:
3225:
3224:
3219:
3217:
3216:
3189:
3187:
3186:
3181:
3179:
3178:
3159:
3157:
3156:
3151:
3128:
3127:
3112:
3111:
3023:
3021:
3020:
3015:
3013:
3009:
2996:
2995:
2974:
2972:
2971:
2966:
2958:
2957:
2935:
2933:
2932:
2927:
2919:
2918:
2890:
2888:
2887:
2882:
2880:
2879:
2768:
2766:
2765:
2760:
2758:
2754:
2753:
2752:
2722:
2721:
2703:
2702:
2685:
2684:
2660:
2658:
2657:
2652:
2647:
2646:
2627:
2625:
2624:
2619:
2614:
2613:
2590:
2588:
2587:
2582:
2580:
2567:
2564:
2559:
2558:
2540:
2539:
2515:
2514:
2496:
2495:
2477:
2476:
2458:
2457:
2439:
2438:
2426:
2425:
2396:
2395:
2383:
2382:
2364:
2363:
2345:
2344:
2326:
2325:
2307:
2306:
2288:
2287:
2275:
2274:
2255:
2231:
2229:
2228:
2223:
2221:
2220:
2196:
2194:
2193:
2188:
2183:
2182:
2170:
2169:
2145:
2144:
2132:
2131:
2113:
2112:
2100:
2099:
2078:
2077:
2065:
2064:
2046:
2045:
2033:
2032:
2020:
2019:
2007:
2006:
1973:
1971:
1970:
1965:
1954:
1953:
1929:
1928:
1901:
1900:
1882:
1881:
1825:
1823:
1822:
1817:
1806:
1805:
1793:
1792:
1774:
1773:
1761:
1760:
1734:
1732:
1731:
1726:
1718:
1717:
1705:
1704:
1686:
1685:
1658:
1657:
1645:
1644:
1626:
1625:
1613:
1612:
1600:
1599:
1587:
1586:
1568:
1567:
1511:random variables
1489:
1487:
1486:
1481:
1461:
1460:
1438:
1436:
1435:
1430:
1410:
1409:
1392:
1390:
1389:
1384:
1364:
1363:
1347:
1345:
1344:
1339:
1337:
1336:
1320:
1318:
1317:
1312:
1310:
1309:
1293:
1291:
1290:
1285:
1265:
1264:
1247:
1245:
1244:
1239:
1209:
1207:
1206:
1201:
1181:
1180:
1160:
1158:
1157:
1152:
1150:
1149:
1129:
1127:
1126:
1121:
1119:
1118:
1099:
1097:
1096:
1091:
1089:
1088:
1072:
1070:
1069:
1064:
1062:
1061:
1045:
1043:
1042:
1037:
1035:
1034:
1018:
1016:
1015:
1010:
999:
998:
982:
980:
979:
974:
972:
971:
955:
953:
952:
947:
936:
935:
912:
910:
909:
904:
899:
885:
884:
865:
863:
862:
857:
849:
848:
832:
828:
826:
825:
820:
818:
817:
564:Continuous-time
533:
532:
341:
334:
327:
117:Elementary event
49:
27:
26:
15925:
15924:
15920:
15919:
15918:
15916:
15915:
15914:
15885:
15884:
15883:
15878:
15860:
15821:Queueing theory
15764:
15706:Skorokhod space
15569:
15560:KunitaâWatanabe
15531:
15497:Sanov's theorem
15467:Ergodic theorem
15440:
15436:Time-reversible
15354:
15317:Queueing models
15311:
15307:SparreâAnderson
15297:CramĂ©râLundberg
15278:
15264:SABR volatility
15170:
15127:
15079:Boolean network
15037:
15023:Renewal process
14954:
14903:Non-homogeneous
14893:Poisson process
14783:Contact process
14746:Brownian motion
14716:Continuous time
14710:
14704:Maximal entropy
14635:
14630:
14600:
14580:
14575:Wayback Machine
14549:
14545:
14540:
14445:J. Laurie Snell
14437:pp. 384ff.
14373:
14344:
14326:Springer-Verlag
14213:
14208:
14207:
14176:
14169:
14159:
14157:
14150:
14144:
14140:
14130:
14128:
14106:
14102:
14073:
14066:
14047:
14040:
14033:
14015:
14011:
13992:
13988:
13975:
13974:
13970:
13961:
13959:
13953:
13949:
13940:
13938:
13927:
13923:
13913:Wayback Machine
13903:
13899:
13886:
13885:
13881:
13865:
13861:
13854:
13840:
13836:
13813:
13809:
13773:10.1.1.225.6090
13744:
13740:
13732:
13721:
13717:
13716:
13712:
13704:
13693:
13687:
13683:
13667:10.1.1.536.8334
13650:
13646:
13617:
13613:
13598:10.2307/1912559
13589:10.1.1.397.3582
13572:
13568:
13552:
13548:
13515:
13511:
13496:
13492:
13477:10.2307/2227127
13471:(250): 318â51.
13461:
13457:
13436:
13432:
13390:
13384:
13380:
13373:
13357:
13353:
13345:
13344:
13340:
13334:Wayback Machine
13324:
13320:
13296:
13292:
13239:
13235:
13213:
13209:
13174:
13170:
13139:
13135:
13104:
13100:
13069:
13065:
13028:
13024:
12993:
12989:
12958:
12954:
12901:
12897:
12842:
12838:
12798:
12794:
12763:
12759:
12728:
12724:
12661:
12657:
12651:
12629:
12625:
12618:
12594:
12590:
12545:
12538:
12529:
12527:
12524:
12518:
12514:
12498:
12489:
12482:
12457:
12453:
12441:
12435:
12431:
12424:
12410:
12406:
12377:
12373:
12366:
12350:
12346:
12337:
12335:
12326:
12325:
12321:
12314:
12298:
12294:
12285:
12283:
12270:
12263:
12256:
12242:
12238:
12228:
12226:
12223:
12214:
12210:
12201:
12199:
12186:
12182:
12175:
12161:
12157:
12148:
12146:
12144:
12125:
12121:
12090:
12086:
12047:
12043:
12034:
12032:
12022:10.1137/1037083
12002:
11998:
11977:
11973:
11964:
11962:
11959:
11955:
11954:
11950:
11943:
11921:
11914:
11907:
11882:
11878:
11871:
11855:
11851:
11838:
11834:
11827:
11811:
11807:
11800:
11784:
11780:
11773:
11751:
11747:
11740:
11724:
11717:
11710:
11694:
11687:
11663:
11659:
11652:
11636:
11632:
11601:
11597:
11590:
11574:
11570:
11563:
11547:
11543:
11514:
11507:
11483:
11474:
11467:
11441:
11437:
11422:10.2307/1403518
11406:
11402:
11387:10.2307/1403785
11368:
11364:
11341:
11337:
11330:
11298:
11294:
11287:
11271:
11264:
11241:
11232:
11225:
11209:
11200:
11193:
11173:
11160:
11151:
11147:
11131:
11127:
11110:
11106:
11089:
11085:
11069:
11065:
11058:
11042:
11038:
11031:
11015:
11011:
11004:
10988:
10984:
10977:
10961:
10957:
10950:
10934:
10927:
10912:
10898:
10894:
10884:
10864:
10860:
10853:
10837:
10833:
10826:
10810:
10806:
10799:
10783:
10776:
10771:
10766:
10737:Master equation
10727:Markov operator
10722:Markov odometer
10677:
10661:
10637:
10616:
10426:
10425:
10407:
10406:
10383:
10382:
10375:
10374:
10323:
10315:Hi Ho! Cherry-O
10307:
10281:to the rise of
10259:
10257:Social sciences
10222:Louis Bachelier
10210:
10190:
10165:
10162:
10161:
10134:
10132:
10130:
10127:
10126:
10099:
10097:
10086:
10082:
10077:
10075:
10072:
10071:
10054:
10050:
10048:
10045:
10044:
10028:
10025:
10024:
10008:
10005:
10004:
9988:
9985:
9984:
9945:
9943:
9932:
9928:
9923:
9921:
9918:
9917:
9910:
9886:Poisson process
9884:according to a
9859:queueing theory
9855:
9853:Queueing theory
9849:
9847:Queueing theory
9776:
9761:
9745:
9733:
9718:Systems biology
9673:
9621:
9620:
9619:
9613:
9575:
9573:
9549:
9542:
9541:
9539:
9531:
9524:
9522:
9521:
9519:
9515:
9513:
9505:
9504:
9502:
9499:
9498:
9494:
9471:
9463:
9431:diffeomorphisms
9403:
9397:
9366:
9360:
9309:
9303:
9298:
9262:
9229:
9215:
9211:
9189:
9185:
9184:
9174:
9170:
9145:
9143:
9135:
9132:
9131:
9124:
9106:
9102:
9094:
9091:
9090:
9073:
9070:
9069:
9053:
9050:
9049:
9018:
9015:
9014:
8995:
8992:
8991:
8977:diagonal matrix
8969:identity matrix
8938:
8917:
8913:
8912:
8898:
8895:
8894:
8867:
8866:
8858:
8856:
8850:
8849:
8835:
8833:
8821:
8817:
8805:
8800:
8791:
8787:
8785:
8778:
8777:
8765:
8761:
8759:
8756:
8755:
8738:
8705:
8698:
8655:
8651:
8642:
8631:
8630:
8629:
8627:
8624:
8623:
8621:
8611:
8592:
8591:
8576: for
8574:
8572:
8560:
8555:
8542:
8538:
8526:
8516:
8515:
8503: for
8501:
8499:
8487:
8482:
8474:
8472:
8469:
8468:
8452:represents the
8436:
8431:
8425:
8422:
8421:
8406:
8393:
8387:
8341:
8340:
8279:
8278:
8261:
8258:
8257:
8235:
8203:
8200:
8199:
8177:
8174:
8173:
8138:
8134:
8132:
8129:
8128:
8112:
8109:
8108:
8076:
8075:
8071:
8063:
8060:
8059:
8033:
8029:
8008:
8004:
7995:
7991:
7983:
7980:
7979:
7951:
7948:
7947:
7930:
7927:
7926:
7908:
7907:
7903:
7895:
7892:
7891:
7884:
7858:
7855:
7854:
7796:
7793:
7792:
7772:
7769:
7768:
7728:
7725:
7724:
7693:
7690:
7689:
7672:
7668:
7666:
7663:
7662:
7628:
7626:
7623:
7622:
7585:
7582:
7581:
7559:
7556:
7555:
7539:
7536:
7535:
7515: and
7513:
7484:
7481:
7480:
7464:
7461:
7460:
7437:
7433:
7416:
7413:
7412:
7392:
7389:
7388:
7351:
7349:
7346:
7345:
7329:
7326:
7325:
7308:
7304:
7302:
7299:
7298:
7282:
7279:
7278:
7267:
7241:
7237:
7235:
7232:
7231:
7215:
7212:
7211:
7193:
7153:
7149:
7147:
7144:
7143:
7127:
7124:
7123:
7105:
7080:
7076:
7065:
7053:
7049:
7047:
7044:
7043:
7037:
6999:
6995:
6993:
6990:
6989:
6952:
6944:
6928:
6917:
6901:
6897:
6882:
6878:
6876:
6873:
6872:
6868:is defined as:
6852:. It is called
6808:
6805:
6804:
6782:
6779:
6778:
6736:
6732:
6717:
6713:
6681:
6678:
6677:
6668:
6662:
6652:
6646:
6640:
6621:
6597:
6582:
6577:
6571:
6561:
6558:
6551:
6527:
6521:
6517:
6512:
6498:
6492:
6488:
6483:
6481:
6478:
6477:
6475:
6468:
6454:
6451:
6445:
6435:
6429:
6426:
6420:is parallel to
6416:
6398:
6397:
6385:
6384:
6379:
6374:
6367:
6355:
6351:
6345:
6341:
6339:
6335:
6334:
6328:
6324:
6308:
6307:
6302:
6297:
6290:
6278:
6274:
6268:
6264:
6262:
6258:
6257:
6251:
6247:
6237:
6236:
6231:
6226:
6219:
6207:
6203:
6197:
6193:
6191:
6187:
6186:
6180:
6176:
6166:
6165:
6160:
6155:
6148:
6144:
6143:
6139:
6133:
6128:
6115:
6114:
6102: for
6100:
6094:
6090:
6081:
6077:
6074:
6067:
6066:
6061:
6056:
6049:
6044:
6034:
6030:
6014:
6013:
6008:
6003:
5996:
5991:
5981:
5977:
5967:
5966:
5961:
5956:
5949:
5944:
5934:
5930:
5921:
5920:
5911:
5906:
5905:
5899:
5891:
5890:
5878:
5877:
5872:
5867:
5860:
5856:
5840:
5839:
5834:
5829:
5822:
5818:
5808:
5807:
5802:
5797:
5790:
5786:
5785:
5781:
5772:
5771:
5762:
5757:
5756:
5750:
5739:
5738:
5729:
5728:
5715:
5704:
5703:
5699:
5683:
5672:
5671:
5667:
5654:
5643:
5642:
5638:
5633:
5626:
5614:
5609:
5608:
5604:
5602:
5599:
5598:
5577:
5571:
5561:
5555:
5545:If we multiply
5524:
5515:
5511:
5501:
5496:
5495:
5489:
5485:
5479:
5468:
5454:
5453:
5448:
5447:
5445:
5442:
5441:
5418:
5413:
5412:
5410:
5407:
5406:
5404:
5387:
5377:
5360:
5348:
5320:
5314:
5310:
5305:
5291:
5285:
5281:
5276:
5268:
5262:
5258:
5253:
5245:
5239:
5235:
5230:
5222:
5219:
5218:
5189:
5178:
5177:
5169:
5167:
5164:
5163:
5155:
5146:
5139:
5132:
5068:
5048:
5043:
5035:
5033:
5030:
5029:
5026:
4999:
4987:has an element
4946:
4942:
4933:
4928:
4927:
4919:
4895:
4890:
4889:
4875:
4873:
4870:
4869:
4797:
4779:identity matrix
4776:
4740:
4735:
4734:
4722:
4717:
4716:
4708:
4700:
4698:
4695:
4694:
4665:
4654:
4652:
4649:
4648:
4623:
4618:
4617:
4605:
4593:
4591:
4588:
4587:
4550:
4549:
4539:
4537:
4527:
4520:
4519:
4509:
4508:
4503:
4497:
4496:
4491:
4481:
4480:
4473:
4472:
4462:
4460:
4450:
4443:
4442:
4440:
4437:
4436:
4419:
4401:
4396:
4395:
4379:
4374:
4373:
4365:
4364:
4359:
4353:
4352:
4347:
4337:
4336:
4328:
4326:
4323:
4322:
4302:
4297:
4296:
4284:
4278:
4275:
4274:
4250:
4245:
4244:
4232:
4226:
4223:
4222:
4192:
4183:
4178:
4177:
4165:
4159:
4156:
4155:
4148:
4140:
4118:
4082:
4078:
4066:
4060:
4057:
4056:
4034:
4030:
4027:
4024:
4023:
3984:
3980:
3974:
3968:
3965:
3964:
3940:
3936:
3935:
3929:
3924:
3919:
3911:
3908:
3907:
3872:
3867:
3864:
3863:
3852:
3849:
3800:
3796:
3775:
3771:
3753:
3749:
3747:
3744:
3743:
3713:
3705:identity matrix
3655:
3653:
3650:
3649:
3636:
3606:
3602:
3587:
3583:
3566:
3562:
3556:
3552:
3551:
3547:
3535:
3531:
3520:
3516:
3515:
3511:
3496:
3492:
3481:
3477:
3476:
3472:
3463:
3459:
3448:
3444:
3443:
3439:
3424:
3420:
3403:
3399:
3398:
3394:
3386:
3383:
3382:
3377:
3370:
3363:
3356:
3349:
3342:
3335:
3319:
3312:
3311:
3303:
3286:
3277:
3270:
3263:
3252:
3242:
3209:
3205:
3203:
3200:
3199:
3192:Kronecker delta
3171:
3167:
3165:
3162:
3161:
3120:
3116:
3104:
3100:
3041:
3038:
3037:
2991:
2987:
2986:
2982:
2980:
2977:
2976:
2947:
2943:
2941:
2938:
2937:
2914:
2910:
2908:
2905:
2904:
2875:
2871:
2869:
2866:
2865:
2854:
2844:
2836:. The elements
2827:
2819:, the elements
2799:
2792:
2782:
2776:
2736:
2732:
2711:
2707:
2698:
2694:
2693:
2689:
2680:
2676:
2674:
2671:
2670:
2642:
2638:
2633:
2630:
2629:
2609:
2605:
2600:
2597:
2596:
2578:
2577:
2565: for
2563:
2548:
2544:
2529:
2525:
2504:
2500:
2485:
2481:
2466:
2462:
2447:
2443:
2434:
2430:
2421:
2417:
2407:
2401:
2400:
2391:
2387:
2378:
2374:
2353:
2349:
2334:
2330:
2315:
2311:
2296:
2292:
2283:
2279:
2270:
2266:
2256:
2254:
2250:
2248:
2245:
2244:
2216:
2212:
2210:
2207:
2206:
2178:
2174:
2159:
2155:
2140:
2136:
2121:
2117:
2108:
2104:
2095:
2091:
2073:
2069:
2060:
2056:
2041:
2037:
2028:
2024:
2015:
2011:
2002:
1998:
1990:
1987:
1986:
1943:
1939:
1924:
1920:
1896:
1892:
1871:
1867:
1859:
1856:
1855:
1851:
1837:
1801:
1797:
1788:
1784:
1769:
1765:
1756:
1752:
1744:
1741:
1740:
1713:
1709:
1700:
1696:
1675:
1671:
1653:
1649:
1640:
1636:
1621:
1617:
1608:
1604:
1595:
1591:
1582:
1578:
1557:
1553:
1545:
1542:
1541:
1535:Markov property
1533:, ... with the
1532:
1525:
1518:
1507:
1501:
1496:
1450:
1446:
1444:
1441:
1440:
1405:
1401:
1399:
1396:
1395:
1359:
1355:
1353:
1350:
1349:
1332:
1328:
1326:
1323:
1322:
1321:now depends on
1305:
1301:
1299:
1296:
1295:
1260:
1256:
1254:
1251:
1250:
1215:
1212:
1211:
1176:
1172:
1170:
1167:
1166:
1145:
1141:
1139:
1136:
1135:
1114:
1110:
1108:
1105:
1104:
1084:
1080:
1078:
1075:
1074:
1057:
1053:
1051:
1048:
1047:
1030:
1026:
1024:
1021:
1020:
994:
990:
988:
985:
984:
967:
963:
961:
958:
957:
931:
927:
925:
922:
921:
895:
880:
876:
871:
868:
867:
844:
840:
838:
835:
834:
830:
813:
809:
807:
804:
803:
799:
778:Poisson process
774:Brownian motion
757:
751:
716:Maurice Fréchet
678:, and proved a
659:Poisson process
651:
635:natural numbers
618:
523:
491:Markov property
483:
467:
453:The adjectives
385:continuous-time
345:
193:Random variable
144:Bernoulli trial
17:
12:
11:
5:
15923:
15913:
15912:
15907:
15902:
15897:
15880:
15879:
15877:
15876:
15871:
15869:List of topics
15865:
15862:
15861:
15859:
15858:
15853:
15848:
15843:
15838:
15833:
15828:
15826:Renewal theory
15823:
15818:
15813:
15808:
15803:
15798:
15793:
15791:Ergodic theory
15788:
15783:
15781:Control theory
15778:
15772:
15770:
15766:
15765:
15763:
15762:
15761:
15760:
15755:
15745:
15740:
15735:
15730:
15725:
15724:
15723:
15713:
15711:Snell envelope
15708:
15703:
15698:
15693:
15688:
15683:
15678:
15673:
15668:
15663:
15658:
15653:
15648:
15643:
15638:
15633:
15628:
15623:
15618:
15613:
15608:
15603:
15598:
15593:
15588:
15583:
15577:
15575:
15571:
15570:
15568:
15567:
15562:
15557:
15552:
15547:
15541:
15539:
15533:
15532:
15530:
15529:
15510:BorelâCantelli
15499:
15494:
15489:
15484:
15479:
15474:
15469:
15464:
15459:
15454:
15448:
15446:
15445:Limit theorems
15442:
15441:
15439:
15438:
15433:
15428:
15423:
15418:
15413:
15408:
15403:
15398:
15393:
15388:
15383:
15378:
15373:
15368:
15362:
15360:
15356:
15355:
15353:
15352:
15347:
15342:
15337:
15332:
15327:
15321:
15319:
15313:
15312:
15310:
15309:
15304:
15299:
15294:
15288:
15286:
15280:
15279:
15277:
15276:
15271:
15266:
15261:
15256:
15251:
15246:
15241:
15236:
15231:
15226:
15221:
15216:
15211:
15206:
15201:
15196:
15191:
15186:
15180:
15178:
15172:
15171:
15169:
15168:
15163:
15158:
15153:
15148:
15143:
15137:
15135:
15129:
15128:
15126:
15125:
15120:
15115:
15114:
15113:
15108:
15098:
15093:
15088:
15083:
15082:
15081:
15076:
15066:
15064:Hopfield model
15061:
15056:
15051:
15045:
15043:
15039:
15038:
15036:
15035:
15030:
15025:
15020:
15015:
15010:
15009:
15008:
15003:
14998:
14993:
14983:
14981:Markov process
14978:
14973:
14968:
14962:
14960:
14956:
14955:
14953:
14952:
14950:Wiener sausage
14947:
14945:Wiener process
14942:
14937:
14932:
14927:
14925:Stable process
14922:
14917:
14915:Semimartingale
14912:
14907:
14906:
14905:
14900:
14890:
14885:
14880:
14875:
14870:
14865:
14860:
14858:Jump diffusion
14855:
14850:
14845:
14840:
14835:
14833:Hawkes process
14830:
14825:
14820:
14815:
14813:Feller process
14810:
14805:
14800:
14795:
14790:
14785:
14780:
14778:Cauchy process
14775:
14774:
14773:
14768:
14763:
14758:
14753:
14743:
14742:
14741:
14731:
14729:Bessel process
14726:
14720:
14718:
14712:
14711:
14709:
14708:
14707:
14706:
14701:
14696:
14691:
14681:
14676:
14671:
14666:
14661:
14656:
14651:
14645:
14643:
14637:
14636:
14629:
14628:
14621:
14614:
14606:
14599:
14598:
14593:
14588:
14577:
14565:
14551:"Markov chain"
14546:
14544:
14543:External links
14541:
14539:
14538:
14523:
14508:
14501:
14483:
14469:
14459:
14441:John G. Kemeny
14438:
14420:
14410:
14392:
14371:
14342:
14306:
14287:
14269:
14251:
14229:
14222:
14214:
14212:
14209:
14206:
14205:
14167:
14138:
14100:
14064:
14038:
14031:
14009:
13994:Kenner, Hugh;
13986:
13968:
13947:
13921:
13897:
13879:
13859:
13852:
13834:
13807:
13738:
13710:
13707:on 2008-12-28.
13681:
13644:
13611:
13566:
13546:
13509:
13490:
13455:
13449:10.1.1.31.1768
13430:
13416:10.1.1.58.8652
13378:
13371:
13351:
13338:
13318:
13304:"Markov chain"
13290:
13233:
13207:
13168:
13133:
13114:(2): 101â112.
13098:
13063:
13022:
12987:
12968:(3): 269â279.
12952:
12895:
12836:
12792:
12757:
12722:
12675:(3): 950â976.
12655:
12649:
12623:
12616:
12588:
12559:(1): 143â154.
12536:
12512:
12487:
12480:
12451:
12429:
12422:
12404:
12391:(1): 295â297.
12371:
12364:
12344:
12319:
12312:
12292:
12275:(1 Dec 2023).
12273:Shalizi, Cosma
12261:
12254:
12236:
12208:
12180:
12173:
12155:
12142:
12119:
12106:(2): 246â290.
12084:
12041:
12016:(3): 387â405.
11996:
11990:10.1.1.28.6191
11971:
11948:
11941:
11912:
11905:
11876:
11869:
11849:
11832:
11825:
11805:
11798:
11778:
11772:978-0471667193
11771:
11745:
11738:
11715:
11708:
11685:
11657:
11650:
11630:
11611:(5): 395â398.
11595:
11588:
11568:
11561:
11541:
11528:(4): 509â546.
11505:
11472:
11465:
11435:
11416:(3): 291â292.
11400:
11381:(3): 255â257.
11362:
11351:(2): 253â268.
11335:
11328:
11311:10.1.1.114.632
11292:
11285:
11262:
11230:
11223:
11198:
11191:
11158:
11145:
11125:
11104:
11094:, Holden-Day.
11083:
11063:
11056:
11036:
11029:
11009:
11002:
10982:
10975:
10955:
10948:
10925:
10910:
10892:
10858:
10851:
10831:
10824:
10804:
10797:
10773:
10772:
10770:
10767:
10765:
10764:
10759:
10754:
10749:
10744:
10739:
10734:
10729:
10724:
10719:
10714:
10709:
10704:
10699:
10694:
10689:
10684:
10678:
10676:
10673:
10660:
10657:
10653:Mark V. Shaney
10636:
10633:
10615:
10612:
10587:
10586:
10583:
10580:
10577:
10573:
10572:
10569:
10566:
10563:
10559:
10558:
10555:
10552:
10549:
10545:
10544:
10541:
10538:
10535:
10531:
10530:
10527:
10524:
10521:
10517:
10516:
10513:
10510:
10507:
10503:
10502:
10499:
10496:
10493:
10489:
10488:
10485:
10482:
10479:
10475:
10474:
10471:
10468:
10465:
10461:
10460:
10457:
10454:
10451:
10440:
10439:
10436:
10433:
10430:
10421:
10420:
10417:
10414:
10411:
10402:
10401:
10398:
10395:
10392:
10388:
10387:
10379:
10371:
10368:
10322:
10319:
10306:
10303:
10263:path-dependent
10258:
10255:
10209:
10206:
10189:
10186:
10169:
10147:
10143:
10140:
10137:
10112:
10108:
10105:
10102:
10096:
10089:
10085:
10081:
10057:
10053:
10032:
10012:
9992:
9958:
9954:
9951:
9948:
9942:
9935:
9931:
9927:
9909:
9906:
9851:Main article:
9848:
9845:
9830:bioinformatics
9780:Claude Shannon
9775:
9772:
9760:
9757:
9744:
9741:
9737:Markov blanket
9732:
9729:
9728:
9727:
9721:
9715:
9709:
9699:
9684:bioinformatics
9672:
9669:
9658:steric effects
9614:
9598:
9581:
9552:
9545:
9534:
9527:
9512:
9497:
9496:
9495:
9493:
9490:
9475:thermodynamics
9470:
9467:
9462:
9459:
9399:Main article:
9396:
9393:
9362:Main article:
9359:
9356:
9353:
9352:
9347:
9342:
9338:
9337:
9332:
9329:
9325:
9324:
9321:
9318:
9305:Main article:
9302:
9299:
9297:
9294:
9251:
9250:
9239:
9232:
9227:
9221:
9218:
9214:
9210:
9207:
9204:
9201:
9198:
9195:
9192:
9188:
9180:
9177:
9173:
9169:
9166:
9163:
9160:
9157:
9154:
9151:
9148:
9142:
9139:
9109:
9105:
9101:
9098:
9077:
9057:
9046:
9045:
9034:
9031:
9028:
9025:
9022:
8999:
8961:
8960:
8949:
8944:
8941:
8936:
8932:
8929:
8926:
8923:
8920:
8916:
8911:
8908:
8905:
8902:
8884:
8883:
8870:
8865:
8857:
8855:
8852:
8851:
8848:
8845:
8842:
8834:
8827:
8824:
8820:
8814:
8811:
8808:
8804:
8797:
8794:
8790:
8784:
8783:
8781:
8776:
8771:
8768:
8764:
8734:
8697:
8694:
8664:
8661:
8658:
8654:
8650:
8645:
8638:
8635:
8617:
8610:
8607:
8606:
8605:
8590:
8587:
8584:
8581:
8573:
8571:
8568:
8563:
8558:
8554:
8548:
8545:
8541:
8535:
8532:
8529:
8525:
8521:
8518:
8517:
8514:
8511:
8508:
8500:
8498:
8495:
8490:
8485:
8481:
8477:
8476:
8454:expected value
8439:
8434:
8430:
8405:
8402:
8389:Main article:
8386:
8383:
8376:autoregressive
8361:
8360:
8349:
8344:
8338:
8335:
8332:
8329:
8326:
8323:
8320:
8317:
8314:
8311:
8308:
8305:
8302:
8299:
8296:
8293:
8290:
8287:
8282:
8277:
8274:
8271:
8268:
8265:
8234:
8231:
8207:
8187:
8184:
8181:
8161:
8158:
8155:
8152:
8149:
8144:
8141:
8137:
8116:
8105:ergodic theory
8079:
8074:
8070:
8067:
8047:
8044:
8041:
8036:
8032:
8028:
8025:
8022:
8019:
8016:
8011:
8007:
8003:
7998:
7994:
7990:
7987:
7967:
7964:
7961:
7958:
7955:
7934:
7911:
7906:
7902:
7899:
7883:
7880:
7879:
7878:
7862:
7842:
7839:
7836:
7833:
7830:
7827:
7824:
7821:
7818:
7815:
7812:
7809:
7806:
7803:
7800:
7776:
7756:
7753:
7750:
7747:
7744:
7741:
7738:
7735:
7732:
7721:
7709:
7706:
7703:
7700:
7697:
7675:
7671:
7650:
7647:
7644:
7640:
7637:
7634:
7631:
7619:
7607:
7604:
7601:
7598:
7595:
7592:
7589:
7569:
7566:
7563:
7543:
7532:
7520:
7512:
7509:
7506:
7503:
7500:
7497:
7494:
7491:
7488:
7468:
7448:
7445:
7440:
7436:
7432:
7429:
7426:
7423:
7420:
7396:
7373:
7370:
7367:
7363:
7360:
7357:
7354:
7333:
7311:
7307:
7286:
7266:
7263:
7244:
7240:
7219:
7201:ergodic theory
7192:
7189:
7169:are positive.
7156:
7152:
7131:
7111:is said to be
7104:
7101:
7088:
7083:
7079:
7075:
7072:
7068:
7064:
7061:
7056:
7052:
7036:
7035:Irreducibility
7033:
7017:null recurrent
7015:is finite and
7002:
6998:
6978:
6977:
6966:
6961:
6958:
6955:
6950:
6947:
6943:
6939:
6936:
6931:
6926:
6923:
6920:
6916:
6912:
6909:
6904:
6900:
6896:
6893:
6890:
6885:
6881:
6840:is said to be
6818:
6815:
6812:
6792:
6789:
6786:
6771:
6770:
6759:
6756:
6753:
6750:
6747:
6744:
6739:
6735:
6731:
6728:
6725:
6720:
6716:
6712:
6709:
6706:
6703:
6700:
6697:
6694:
6691:
6688:
6685:
6620:
6617:
6596:
6593:
6581:
6578:
6573:Main article:
6570:
6567:
6556:
6549:
6534:
6530:
6524:
6520:
6515:
6511:
6508:
6505:
6501:
6495:
6491:
6486:
6473:
6466:
6449:
6443:
6439:approaches to
6424:
6412:
6411:
6395:
6388:
6382:
6377:
6370:
6365:
6358:
6354:
6348:
6344:
6338:
6331:
6327:
6323:
6320:
6317:
6311:
6305:
6300:
6293:
6288:
6281:
6277:
6271:
6267:
6261:
6254:
6250:
6246:
6240:
6234:
6229:
6222:
6217:
6210:
6206:
6200:
6196:
6190:
6183:
6179:
6175:
6169:
6163:
6158:
6151:
6147:
6142:
6136:
6131:
6127:
6123:
6120:
6118:
6116:
6113:
6110:
6107:
6097:
6093:
6089:
6084:
6080:
6076:
6070:
6064:
6059:
6052:
6047:
6043:
6037:
6033:
6029:
6026:
6023:
6017:
6011:
6006:
5999:
5994:
5990:
5984:
5980:
5976:
5970:
5964:
5959:
5952:
5947:
5943:
5937:
5933:
5929:
5926:
5924:
5922:
5917:
5914:
5909:
5902:
5897:
5894:
5888:
5881:
5875:
5870:
5863:
5859:
5855:
5852:
5849:
5843:
5837:
5832:
5825:
5821:
5817:
5811:
5805:
5800:
5793:
5789:
5784:
5780:
5777:
5775:
5773:
5768:
5765:
5760:
5753:
5748:
5745:
5742:
5737:
5734:
5732:
5730:
5726:
5721:
5718:
5713:
5710:
5707:
5702:
5698:
5694:
5689:
5686:
5681:
5678:
5675:
5670:
5665:
5660:
5657:
5652:
5649:
5646:
5641:
5636:
5632:
5629:
5627:
5623:
5620:
5617:
5612:
5607:
5606:
5575:
5569:
5543:
5542:
5531:
5527:
5523:
5518:
5514:
5509:
5504:
5499:
5492:
5488:
5482:
5477:
5474:
5471:
5467:
5463:
5457:
5451:
5426:
5421:
5416:
5400:
5383:
5373:
5365:-th column of
5356:
5339:
5338:
5327:
5323:
5317:
5313:
5308:
5304:
5301:
5298:
5294:
5288:
5284:
5279:
5275:
5271:
5265:
5261:
5256:
5252:
5248:
5242:
5238:
5233:
5229:
5226:
5212:
5211:
5200:
5195:
5192:
5187:
5184:
5181:
5176:
5172:
5151:
5144:
5137:
5130:
5055:
5051:
5046:
5042:
5038:
5025:
5022:
4991:
4981:
4980:
4968:
4957:
4952:
4949:
4945:
4941:
4936:
4931:
4926:
4922:
4918:
4915:
4912:
4909:
4904:
4901:
4898:
4893:
4888:
4885:
4882:
4878:
4789:
4772:
4766:
4765:
4754:
4749:
4746:
4743:
4738:
4733:
4730:
4725:
4720:
4715:
4711:
4707:
4703:
4684:
4683:
4672:
4668:
4664:
4660:
4657:
4631:
4626:
4621:
4614:
4611:
4608:
4604:
4600:
4596:
4568:
4567:
4554:
4546:
4543:
4538:
4534:
4531:
4526:
4525:
4523:
4518:
4513:
4507:
4504:
4502:
4499:
4498:
4495:
4492:
4490:
4487:
4486:
4484:
4477:
4469:
4466:
4461:
4457:
4454:
4449:
4448:
4446:
4434:
4422:
4418:
4413:
4410:
4407:
4404:
4399:
4393:
4390:
4385:
4382:
4377:
4369:
4363:
4360:
4358:
4355:
4354:
4351:
4348:
4346:
4343:
4342:
4340:
4335:
4331:
4305:
4300:
4293:
4290:
4287:
4283:
4253:
4248:
4241:
4238:
4235:
4231:
4211:
4210:
4199:
4195:
4191:
4186:
4181:
4174:
4171:
4168:
4164:
4117:
4114:
4093:
4090:
4085:
4081:
4077:
4074:
4069:
4065:
4037:
4033:
3995:
3992:
3987:
3983:
3977:
3973:
3961:
3960:
3943:
3939:
3932:
3928:
3923:
3918:
3915:
3897:
3896:
3885:
3882:
3879:
3875:
3871:
3848:
3845:
3829:
3828:
3817:
3814:
3811:
3808:
3803:
3799:
3795:
3792:
3789:
3784:
3781:
3778:
3774:
3770:
3767:
3764:
3759:
3756:
3752:
3712:
3709:
3701:
3700:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3661:
3658:
3632:
3626:
3625:
3614:
3609:
3605:
3601:
3596:
3593:
3590:
3586:
3582:
3575:
3572:
3569:
3565:
3559:
3555:
3550:
3546:
3543:
3538:
3534:
3530:
3523:
3519:
3514:
3510:
3507:
3504:
3499:
3495:
3491:
3484:
3480:
3475:
3471:
3466:
3462:
3458:
3451:
3447:
3442:
3438:
3433:
3430:
3427:
3423:
3419:
3412:
3409:
3406:
3402:
3397:
3393:
3390:
3375:
3368:
3361:
3354:
3347:
3340:
3333:
3321:For any value
3318:
3315:
3307:
3299:
3295:
3282:
3275:
3268:
3261:
3248:
3241:
3238:
3215:
3212:
3208:
3177:
3174:
3170:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3126:
3123:
3119:
3115:
3110:
3107:
3103:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3012:
3008:
3005:
3002:
2999:
2994:
2990:
2985:
2964:
2961:
2956:
2953:
2950:
2946:
2925:
2922:
2917:
2913:
2878:
2874:
2853:
2850:
2840:
2823:
2798: â„ 0
2794:
2788:
2778:Main article:
2775:
2772:
2771:
2770:
2757:
2751:
2748:
2745:
2742:
2739:
2735:
2731:
2728:
2725:
2720:
2717:
2714:
2710:
2706:
2701:
2697:
2692:
2688:
2683:
2679:
2669:values, i.e.,
2650:
2645:
2641:
2637:
2617:
2612:
2608:
2604:
2576:
2573:
2570:
2562:
2557:
2554:
2551:
2547:
2543:
2538:
2535:
2532:
2528:
2524:
2521:
2518:
2513:
2510:
2507:
2503:
2499:
2494:
2491:
2488:
2484:
2480:
2475:
2472:
2469:
2465:
2461:
2456:
2453:
2450:
2446:
2442:
2437:
2433:
2429:
2424:
2420:
2416:
2413:
2410:
2408:
2406:
2403:
2402:
2399:
2394:
2390:
2386:
2381:
2377:
2373:
2370:
2367:
2362:
2359:
2356:
2352:
2348:
2343:
2340:
2337:
2333:
2329:
2324:
2321:
2318:
2314:
2310:
2305:
2302:
2299:
2295:
2291:
2286:
2282:
2278:
2273:
2269:
2265:
2262:
2259:
2257:
2253:
2252:
2233:
2219:
2215:
2186:
2181:
2177:
2173:
2168:
2165:
2162:
2158:
2154:
2151:
2148:
2143:
2139:
2135:
2130:
2127:
2124:
2120:
2116:
2111:
2107:
2103:
2098:
2094:
2090:
2087:
2084:
2081:
2076:
2072:
2068:
2063:
2059:
2055:
2052:
2049:
2044:
2040:
2036:
2031:
2027:
2023:
2018:
2014:
2010:
2005:
2001:
1997:
1994:
1983:
1963:
1960:
1957:
1952:
1949:
1946:
1942:
1938:
1935:
1932:
1927:
1923:
1919:
1916:
1913:
1910:
1907:
1904:
1899:
1895:
1891:
1888:
1885:
1880:
1877:
1874:
1870:
1866:
1863:
1850:
1847:
1833:
1827:
1826:
1815:
1812:
1809:
1804:
1800:
1796:
1791:
1787:
1783:
1780:
1777:
1772:
1768:
1764:
1759:
1755:
1751:
1748:
1724:
1721:
1716:
1712:
1708:
1703:
1699:
1695:
1692:
1689:
1684:
1681:
1678:
1674:
1670:
1667:
1664:
1661:
1656:
1652:
1648:
1643:
1639:
1635:
1632:
1629:
1624:
1620:
1616:
1611:
1607:
1603:
1598:
1594:
1590:
1585:
1581:
1577:
1574:
1571:
1566:
1563:
1560:
1556:
1552:
1549:
1530:
1523:
1516:
1503:Main article:
1500:
1497:
1495:
1492:
1479:
1476:
1473:
1470:
1467:
1464:
1459:
1456:
1453:
1449:
1428:
1425:
1422:
1419:
1416:
1413:
1408:
1404:
1382:
1379:
1376:
1373:
1370:
1367:
1362:
1358:
1335:
1331:
1308:
1304:
1283:
1280:
1277:
1274:
1271:
1268:
1263:
1259:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1199:
1196:
1193:
1190:
1187:
1184:
1179:
1175:
1148:
1144:
1117:
1113:
1087:
1083:
1060:
1056:
1033:
1029:
1008:
1005:
1002:
997:
993:
970:
966:
945:
942:
939:
934:
930:
902:
898:
894:
891:
888:
883:
879:
875:
855:
852:
847:
843:
816:
812:
798:
795:
794:
793:
789:
781:
770:Wiener process
765:gambler's ruin
753:Main article:
750:
747:
739:William Feller
731:Sydney Chapman
727:Norbert Wiener
703:Francis Galton
687:Henri Poincaré
663:Pavel Nekrasov
650:
647:
617:
614:
580:
579:
576:Wiener process
568:
565:
561:
560:
555:(for example,
550:
547:
546:Discrete-time
543:
542:
539:
536:
522:
519:
495:memorylessness
482:
479:
466:
463:
395:mathematician
357:Markov process
347:
346:
344:
343:
336:
329:
321:
318:
317:
316:
315:
310:
302:
301:
300:
299:
294:
292:Bayes' theorem
289:
284:
279:
274:
266:
265:
264:
263:
258:
253:
248:
240:
239:
238:
237:
236:
235:
230:
225:
223:Observed value
220:
215:
210:
208:Expected value
205:
200:
190:
185:
184:
183:
178:
173:
168:
163:
158:
148:
147:
146:
136:
135:
134:
129:
124:
119:
114:
104:
99:
91:
90:
89:
88:
83:
78:
77:
76:
66:
65:
64:
51:
50:
42:
41:
35:
34:
15:
9:
6:
4:
3:
2:
15922:
15911:
15908:
15906:
15903:
15901:
15900:Markov models
15898:
15896:
15893:
15892:
15890:
15875:
15872:
15870:
15867:
15866:
15863:
15857:
15854:
15852:
15849:
15847:
15844:
15842:
15839:
15837:
15834:
15832:
15829:
15827:
15824:
15822:
15819:
15817:
15814:
15812:
15809:
15807:
15804:
15802:
15799:
15797:
15794:
15792:
15789:
15787:
15784:
15782:
15779:
15777:
15774:
15773:
15771:
15767:
15759:
15756:
15754:
15751:
15750:
15749:
15746:
15744:
15741:
15739:
15736:
15734:
15731:
15729:
15728:Stopping time
15726:
15722:
15719:
15718:
15717:
15714:
15712:
15709:
15707:
15704:
15702:
15699:
15697:
15694:
15692:
15689:
15687:
15684:
15682:
15679:
15677:
15674:
15672:
15669:
15667:
15664:
15662:
15659:
15657:
15654:
15652:
15649:
15647:
15644:
15642:
15639:
15637:
15634:
15632:
15629:
15627:
15624:
15622:
15619:
15617:
15614:
15612:
15609:
15607:
15604:
15602:
15599:
15597:
15594:
15592:
15589:
15587:
15584:
15582:
15579:
15578:
15576:
15572:
15566:
15563:
15561:
15558:
15556:
15553:
15551:
15548:
15546:
15543:
15542:
15540:
15538:
15534:
15527:
15523:
15519:
15518:HewittâSavage
15515:
15511:
15507:
15503:
15502:Zeroâone laws
15500:
15498:
15495:
15493:
15490:
15488:
15485:
15483:
15480:
15478:
15475:
15473:
15470:
15468:
15465:
15463:
15460:
15458:
15455:
15453:
15450:
15449:
15447:
15443:
15437:
15434:
15432:
15429:
15427:
15424:
15422:
15419:
15417:
15414:
15412:
15409:
15407:
15404:
15402:
15399:
15397:
15394:
15392:
15389:
15387:
15384:
15382:
15379:
15377:
15374:
15372:
15369:
15367:
15364:
15363:
15361:
15357:
15351:
15348:
15346:
15343:
15341:
15338:
15336:
15333:
15331:
15328:
15326:
15323:
15322:
15320:
15318:
15314:
15308:
15305:
15303:
15300:
15298:
15295:
15293:
15290:
15289:
15287:
15285:
15281:
15275:
15272:
15270:
15267:
15265:
15262:
15260:
15257:
15255:
15252:
15250:
15247:
15245:
15242:
15240:
15237:
15235:
15232:
15230:
15227:
15225:
15222:
15220:
15217:
15215:
15212:
15210:
15207:
15205:
15202:
15200:
15199:BlackâScholes
15197:
15195:
15192:
15190:
15187:
15185:
15182:
15181:
15179:
15177:
15173:
15167:
15164:
15162:
15159:
15157:
15154:
15152:
15149:
15147:
15144:
15142:
15139:
15138:
15136:
15134:
15130:
15124:
15121:
15119:
15116:
15112:
15109:
15107:
15104:
15103:
15102:
15101:Point process
15099:
15097:
15094:
15092:
15089:
15087:
15084:
15080:
15077:
15075:
15072:
15071:
15070:
15067:
15065:
15062:
15060:
15059:Gibbs measure
15057:
15055:
15052:
15050:
15047:
15046:
15044:
15040:
15034:
15031:
15029:
15026:
15024:
15021:
15019:
15016:
15014:
15011:
15007:
15004:
15002:
14999:
14997:
14994:
14992:
14989:
14988:
14987:
14984:
14982:
14979:
14977:
14974:
14972:
14969:
14967:
14964:
14963:
14961:
14957:
14951:
14948:
14946:
14943:
14941:
14938:
14936:
14933:
14931:
14928:
14926:
14923:
14921:
14918:
14916:
14913:
14911:
14908:
14904:
14901:
14899:
14896:
14895:
14894:
14891:
14889:
14886:
14884:
14881:
14879:
14876:
14874:
14871:
14869:
14866:
14864:
14861:
14859:
14856:
14854:
14851:
14849:
14848:ItĂŽ diffusion
14846:
14844:
14841:
14839:
14836:
14834:
14831:
14829:
14826:
14824:
14823:Gamma process
14821:
14819:
14816:
14814:
14811:
14809:
14806:
14804:
14801:
14799:
14796:
14794:
14791:
14789:
14786:
14784:
14781:
14779:
14776:
14772:
14769:
14767:
14764:
14762:
14759:
14757:
14754:
14752:
14749:
14748:
14747:
14744:
14740:
14737:
14736:
14735:
14732:
14730:
14727:
14725:
14722:
14721:
14719:
14717:
14713:
14705:
14702:
14700:
14697:
14695:
14694:Self-avoiding
14692:
14690:
14687:
14686:
14685:
14682:
14680:
14679:Moran process
14677:
14675:
14672:
14670:
14667:
14665:
14662:
14660:
14657:
14655:
14652:
14650:
14647:
14646:
14644:
14642:
14641:Discrete time
14638:
14634:
14627:
14622:
14620:
14615:
14613:
14608:
14607:
14604:
14597:
14594:
14592:
14589:
14587:
14583:
14578:
14576:
14572:
14569:
14566:
14562:
14558:
14557:
14552:
14548:
14547:
14536:
14532:
14528:
14524:
14521:
14520:0-7923-9650-2
14517:
14513:
14509:
14506:
14502:
14499:
14498:0-471-33341-7
14495:
14491:
14487:
14484:
14482:
14478:
14474:
14470:
14468:
14467:0-521-60494-X
14464:
14460:
14458:
14457:0-442-04328-7
14454:
14450:
14446:
14442:
14439:
14436:
14429:
14428:
14421:
14416:
14411:
14409:
14405:
14401:
14397:
14393:
14390:
14386:
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14364:
14360:
14355:
14354:
14345:
14339:
14335:
14331:
14327:
14323:
14318:
14317:
14311:
14307:
14304:
14300:
14299:0-387-19832-6
14296:
14292:
14288:
14285:
14284:0-471-52369-0
14281:
14277:
14273:
14270:
14267:
14266:0-89871-296-3
14263:
14260:
14256:
14252:
14248:
14244:
14240:
14236:
14230:
14227:
14223:
14220:
14216:
14215:
14201:
14197:
14193:
14189:
14185:
14181:
14174:
14172:
14156:
14149:
14142:
14127:
14123:
14119:
14115:
14111:
14104:
14095:
14090:
14086:
14082:
14078:
14071:
14069:
14060:
14056:
14052:
14045:
14043:
14034:
14028:
14023:
14022:
14013:
14005:
14001:
13997:
13990:
13982:
13978:
13972:
13958:
13951:
13937:on 2007-12-09
13936:
13932:
13925:
13918:
13914:
13910:
13907:
13901:
13893:
13889:
13888:"Continuator"
13883:
13877:
13873:
13869:
13863:
13855:
13849:
13846:. MIT Press.
13845:
13838:
13830:
13826:
13822:
13818:
13811:
13803:
13799:
13794:
13789:
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13779:
13774:
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13753:
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13742:
13731:
13727:
13720:
13714:
13703:
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13677:
13673:
13668:
13663:
13659:
13655:
13648:
13640:
13636:
13632:
13628:
13627:
13622:
13615:
13607:
13603:
13599:
13595:
13590:
13585:
13582:(2): 357â84.
13581:
13577:
13570:
13562:
13558:
13550:
13541:
13536:
13532:
13528:
13524:
13520:
13513:
13505:
13501:
13500:Am. Econ. Rev
13494:
13486:
13482:
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13470:
13466:
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13441:
13434:
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13311:
13310:
13305:
13301:
13294:
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13282:
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13260:
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13187:
13183:
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13144:
13137:
13129:
13125:
13121:
13117:
13113:
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13094:
13090:
13086:
13082:
13078:
13074:
13067:
13058:
13053:
13049:
13045:
13041:
13037:
13033:
13026:
13018:
13014:
13010:
13006:
13002:
12998:
12991:
12983:
12979:
12975:
12971:
12967:
12963:
12956:
12948:
12944:
12939:
12934:
12930:
12926:
12922:
12918:
12914:
12910:
12909:AIChE Journal
12906:
12899:
12891:
12887:
12882:
12877:
12872:
12867:
12863:
12859:
12855:
12851:
12847:
12840:
12832:
12828:
12824:
12820:
12816:
12812:
12808:
12804:
12796:
12788:
12784:
12780:
12776:
12772:
12768:
12761:
12753:
12749:
12745:
12741:
12737:
12733:
12726:
12718:
12714:
12709:
12704:
12700:
12696:
12692:
12688:
12683:
12678:
12674:
12670:
12666:
12659:
12652:
12650:9781441967657
12646:
12642:
12638:
12634:
12627:
12619:
12613:
12609:
12605:
12601:
12600:
12592:
12584:
12580:
12575:
12570:
12566:
12562:
12558:
12554:
12550:
12543:
12541:
12523:
12516:
12510:
12506:
12502:
12496:
12494:
12492:
12483:
12481:9780511810633
12477:
12473:
12469:
12465:
12464:Markov Chains
12461:
12460:Norris, J. R.
12455:
12447:
12440:
12433:
12425:
12419:
12415:
12408:
12399:
12394:
12390:
12386:
12382:
12375:
12367:
12361:
12357:
12356:
12348:
12333:
12329:
12323:
12315:
12309:
12305:
12304:
12296:
12282:
12278:
12274:
12268:
12266:
12257:
12255:0-8162-6664-6
12251:
12247:
12240:
12222:
12218:
12217:Lalley, Steve
12212:
12198:
12194:
12190:
12184:
12176:
12174:0-07-028631-0
12170:
12166:
12159:
12145:
12143:9780719022067
12139:
12135:
12134:
12129:
12123:
12114:
12109:
12105:
12101:
12100:
12095:
12088:
12080:
12076:
12072:
12068:
12064:
12060:
12057:(4): 041112.
12056:
12052:
12045:
12031:
12027:
12023:
12019:
12015:
12011:
12007:
12000:
11991:
11986:
11982:
11975:
11958:
11952:
11944:
11938:
11934:
11930:
11926:
11919:
11917:
11908:
11906:9780511810633
11902:
11898:
11894:
11890:
11889:Markov Chains
11886:
11885:Norris, J. R.
11880:
11872:
11866:
11862:
11861:
11853:
11845:
11844:
11836:
11828:
11822:
11818:
11817:
11809:
11801:
11795:
11791:
11790:
11782:
11774:
11768:
11764:
11760:
11757:. p. 1.
11756:
11749:
11741:
11735:
11731:
11730:
11722:
11720:
11711:
11705:
11701:
11700:
11692:
11690:
11681:
11677:
11673:
11669:
11661:
11653:
11647:
11643:
11642:
11634:
11626:
11622:
11618:
11614:
11610:
11606:
11599:
11591:
11585:
11581:
11580:
11572:
11564:
11558:
11554:
11553:
11545:
11536:
11531:
11527:
11523:
11519:
11512:
11510:
11501:
11497:
11493:
11489:
11481:
11479:
11477:
11468:
11462:
11458:
11454:
11450:
11446:
11439:
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11404:
11396:
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11325:
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11207:
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11194:
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11184:
11180:
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11171:
11169:
11167:
11165:
11163:
11155:
11149:
11143:
11142:0-19-920613-9
11139:
11135:
11129:
11122:
11121:0-19-920613-9
11118:
11114:
11108:
11101:
11100:0-8162-6664-6
11097:
11093:
11087:
11081:
11080:0-521-81099-X
11077:
11073:
11067:
11059:
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11049:
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11022:
11021:
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10972:
10968:
10967:
10959:
10951:
10945:
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10932:
10930:
10921:
10917:
10913:
10907:
10903:
10896:
10888:
10880:
10876:
10875:
10869:
10862:
10854:
10848:
10845:. CRC Press.
10844:
10843:
10835:
10827:
10821:
10817:
10816:
10808:
10800:
10794:
10790:
10789:
10781:
10779:
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10720:
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10715:
10713:
10710:
10708:
10705:
10703:
10700:
10698:
10695:
10693:
10690:
10688:
10685:
10683:
10680:
10679:
10672:
10670:
10666:
10656:
10654:
10650:
10646:
10642:
10632:
10630:
10626:
10625:base stealing
10622:
10611:
10607:
10604:
10602:
10598:
10594:
10584:
10581:
10578:
10575:
10574:
10570:
10567:
10564:
10561:
10560:
10556:
10553:
10550:
10547:
10546:
10542:
10539:
10536:
10533:
10532:
10528:
10525:
10522:
10519:
10518:
10514:
10511:
10508:
10505:
10504:
10500:
10497:
10494:
10491:
10490:
10486:
10483:
10480:
10477:
10476:
10472:
10469:
10466:
10463:
10462:
10458:
10455:
10452:
10449:
10448:
10437:
10434:
10431:
10423:
10422:
10418:
10415:
10412:
10404:
10403:
10399:
10396:
10393:
10390:
10389:
10380:
10372:
10369:
10366:
10365:
10359:
10357:
10353:
10348:
10344:
10343:SuperCollider
10340:
10336:
10332:
10328:
10318:
10316:
10312:
10302:
10300:
10296:
10295:authoritarian
10292:
10288:
10284:
10280:
10275:
10274:
10268:
10264:
10254:
10252:
10248:
10246:
10241:
10239:
10235:
10231:
10227:
10223:
10219:
10215:
10205:
10203:
10199:
10195:
10185:
10181:
10167:
10145:
10141:
10138:
10135:
10110:
10106:
10103:
10100:
10094:
10087:
10083:
10079:
10055:
10051:
10030:
10010:
9990:
9982:
9978:
9956:
9952:
9949:
9946:
9940:
9933:
9929:
9925:
9914:
9905:
9903:
9899:
9895:
9891:
9887:
9883:
9879:
9875:
9871:
9866:
9864:
9860:
9854:
9844:
9842:
9838:
9833:
9831:
9827:
9822:
9820:
9816:
9812:
9808:
9804:
9800:
9795:
9791:
9787:
9786:
9781:
9771:
9769:
9765:
9756:
9753:
9749:
9740:
9738:
9725:
9722:
9719:
9716:
9713:
9710:
9707:
9703:
9700:
9697:
9693:
9689:
9686:, where most
9685:
9681:
9680:Phylogenetics
9678:
9677:
9676:
9668:
9666:
9661:
9659:
9654:
9649:
9646:
9641:
9639:
9634:
9632:
9627:
9617:
9596:
9543:
9532:
9510:
9489:
9488:simulations.
9487:
9482:
9480:
9476:
9466:
9458:
9456:
9452:
9448:
9447:sofic systems
9444:
9443:Chacon system
9440:
9436:
9432:
9428:
9424:
9420:
9416:
9412:
9408:
9402:
9392:
9390:
9387:
9383:
9378:
9376:
9371:
9365:
9351:
9348:
9346:
9343:
9340:
9339:
9336:
9333:
9331:Markov chain
9330:
9327:
9326:
9322:
9319:
9317:
9316:
9313:
9308:
9293:
9291:
9287:
9283:
9279:
9275:
9270:
9268:
9261:is not. Once
9260:
9256:
9237:
9230:
9219:
9216:
9205:
9199:
9196:
9190:
9178:
9175:
9164:
9158:
9155:
9149:
9146:
9140:
9137:
9130:
9129:
9128:
9122:
9107:
9099:
9075:
9055:
9032:
9029:
9026:
9023:
9020:
9013:
9012:
9011:
8997:
8988:
8986:
8982:
8981:main diagonal
8978:
8974:
8970:
8966:
8947:
8942:
8939:
8934:
8927:
8921:
8918:
8914:
8909:
8906:
8903:
8900:
8893:
8892:
8891:
8889:
8863:
8853:
8846:
8843:
8840:
8825:
8822:
8818:
8812:
8809:
8806:
8802:
8795:
8792:
8788:
8779:
8774:
8769:
8766:
8762:
8754:
8753:
8752:
8750:
8746:
8742:
8737:
8733:
8729:
8725:
8724:
8719:
8715:
8711:
8703:
8693:
8691:
8687:
8682:
8680:
8679:Kelly's lemma
8662:
8659:
8656:
8652:
8648:
8643:
8633:
8620:
8616:
8609:Time reversal
8588:
8585:
8582:
8579:
8569:
8566:
8561:
8556:
8552:
8546:
8543:
8539:
8533:
8530:
8527:
8523:
8519:
8512:
8509:
8506:
8496:
8493:
8488:
8483:
8479:
8467:
8466:
8465:
8463:
8459:
8455:
8437:
8432:
8428:
8419:
8416:, the vector
8415:
8412: â
8411:
8401:
8398:
8392:
8385:Hitting times
8382:
8380:
8377:
8372:
8370:
8366:
8347:
8330:
8324:
8321:
8315:
8309:
8303:
8300:
8297:
8291:
8285:
8275:
8269:
8263:
8256:
8255:
8254:
8252:
8248:
8244:
8240:
8230:
8228:
8224:
8219:
8182:
8179:
8159:
8156:
8150:
8142:
8139:
8135:
8114:
8106:
8101:
8099:
8094:
8068:
8042:
8039:
8034:
8030:
8023:
8017:
8014:
8009:
8005:
8001:
7996:
7992:
7985:
7956:
7953:
7900:
7889:
7876:
7860:
7837:
7834:
7831:
7828:
7825:
7819:
7816:
7810:
7807:
7804:
7798:
7790:
7774:
7751:
7748:
7745:
7739:
7736:
7733:
7730:
7722:
7707:
7704:
7701:
7698:
7695:
7673:
7669:
7645:
7620:
7605:
7602:
7599:
7596:
7593:
7590:
7587:
7567:
7564:
7561:
7541:
7533:
7518:
7510:
7504:
7498:
7492:
7486:
7466:
7446:
7443:
7438:
7430:
7427:
7424:
7418:
7410:
7409:
7408:
7394:
7385:
7368:
7331:
7309:
7305:
7284:
7276:
7272:
7262:
7260:
7242:
7238:
7217:
7209:
7204:
7202:
7198:
7188:
7185:
7183:
7179:
7175:
7170:
7154:
7150:
7129:
7120:
7118:
7114:
7110:
7100:
7081:
7077:
7070:
7066:
7062:
7059:
7054:
7050:
7040:
7032:
7030:
7026:
7021:
7018:
7000:
6996:
6987:
6983:
6964:
6956:
6948:
6945:
6941:
6937:
6934:
6924:
6921:
6918:
6914:
6910:
6902:
6898:
6891:
6888:
6883:
6879:
6871:
6870:
6869:
6867:
6863:
6859:
6855:
6851:
6847:
6843:
6839:
6834:
6832:
6816:
6813:
6810:
6790:
6787:
6784:
6776:
6773:The state is
6754:
6751:
6745:
6742:
6737:
6733:
6729:
6726:
6723:
6718:
6714:
6704:
6701:
6698:
6695:
6686:
6683:
6676:
6675:
6674:
6671:
6665:
6660:
6655:
6649:
6643:
6637:
6635:
6631:
6626:
6616:
6614:
6610:
6606:
6602:
6592:
6589:
6587:
6586:Harris chains
6580:Harris chains
6576:
6566:
6564:
6555:
6548:
6532:
6522:
6518:
6509:
6506:
6503:
6493:
6489:
6472:
6465:
6461:
6457:
6448:
6442:
6438:
6432:
6423:
6419:
6393:
6380:
6368:
6363:
6356:
6352:
6346:
6342:
6336:
6329:
6325:
6321:
6318:
6315:
6303:
6291:
6286:
6279:
6275:
6269:
6265:
6259:
6252:
6248:
6244:
6232:
6220:
6215:
6208:
6204:
6198:
6194:
6188:
6181:
6177:
6173:
6161:
6149:
6145:
6140:
6134:
6129:
6125:
6121:
6119:
6111:
6108:
6105:
6095:
6091:
6082:
6078:
6062:
6050:
6045:
6041:
6035:
6031:
6027:
6024:
6021:
6009:
5997:
5992:
5988:
5982:
5978:
5974:
5962:
5950:
5945:
5941:
5935:
5931:
5927:
5925:
5915:
5912:
5900:
5886:
5873:
5861:
5857:
5853:
5850:
5847:
5835:
5823:
5819:
5815:
5803:
5791:
5787:
5782:
5778:
5776:
5766:
5763:
5751:
5735:
5733:
5724:
5719:
5716:
5700:
5696:
5692:
5687:
5684:
5668:
5663:
5658:
5655:
5639:
5630:
5628:
5618:
5597:
5596:
5595:
5593:
5589:
5585:
5581:
5574:
5568:
5564:
5558:
5552:
5548:
5529:
5521:
5516:
5512:
5507:
5502:
5490:
5486:
5480:
5475:
5472:
5469:
5465:
5461:
5440:
5439:
5438:
5437:we can write
5424:
5419:
5403:
5399:
5395:
5391:
5386:
5381:
5376:
5372:
5368:
5364:
5359:
5355:
5351:
5344:
5325:
5315:
5311:
5302:
5299:
5296:
5286:
5282:
5273:
5263:
5259:
5250:
5240:
5236:
5227:
5224:
5217:
5216:
5215:
5198:
5193:
5190:
5174:
5162:
5161:
5160:
5159:
5154:
5150:
5143:
5136:
5129:
5125:
5121:
5117:
5113:
5109:
5104:
5102:
5098:
5094:
5090:
5086:
5082:
5078:
5075:
5071:
5053:
5040:
5021:
5019:
5015:
5011:
5008:. Hence, the
5007:
5003:
4998:
4994:
4990:
4986:
4977:
4973:
4969:
4955:
4950:
4947:
4934:
4924:
4913:
4902:
4899:
4896:
4883:
4880:
4868:
4867:
4866:
4864:
4860:
4856:
4851:
4849:
4845:
4841:
4837:
4833:
4829:
4825:
4821:
4817:
4813:
4809:
4805:
4801:
4796:
4792:
4788:
4784:
4780:
4775:
4771:
4752:
4747:
4744:
4741:
4731:
4723:
4713:
4693:
4692:
4691:
4689:
4670:
4662:
4647:
4646:
4645:
4642:
4629:
4624:
4606:
4598:
4585:
4581:
4577:
4571:
4552:
4544:
4541:
4532:
4529:
4521:
4516:
4511:
4505:
4500:
4493:
4488:
4482:
4475:
4467:
4464:
4455:
4452:
4444:
4435:
4416:
4411:
4408:
4405:
4402:
4391:
4388:
4383:
4380:
4367:
4361:
4356:
4349:
4344:
4338:
4333:
4321:
4320:
4319:
4303:
4285:
4272:
4267:
4251:
4233:
4220:
4216:
4197:
4189:
4184:
4166:
4154:
4153:
4152:
4146:
4137:
4135:
4131:
4127:
4123:
4113:
4111:
4107:
4091:
4088:
4083:
4079:
4075:
4072:
4067:
4063:
4054:
4035:
4031:
4020:
4017:
4013:
4009:
3993:
3990:
3985:
3981:
3975:
3971:
3941:
3937:
3930:
3926:
3921:
3916:
3913:
3906:
3905:
3904:
3902:
3883:
3880:
3877:
3869:
3862:
3861:
3860:
3858:
3844:
3842:
3838:
3834:
3815:
3809:
3806:
3801:
3797:
3793:
3790:
3787:
3782:
3779:
3776:
3772:
3762:
3757:
3754:
3750:
3742:
3741:
3740:
3738:
3734:
3730:
3726:
3722:
3718:
3708:
3706:
3687:
3681:
3675:
3672:
3666:
3659:
3656:
3648:
3647:
3646:
3644:
3640:
3635:
3631:
3607:
3603:
3599:
3594:
3591:
3588:
3584:
3573:
3570:
3567:
3563:
3557:
3553:
3548:
3544:
3536:
3532:
3528:
3521:
3517:
3512:
3508:
3505:
3502:
3497:
3493:
3489:
3482:
3478:
3473:
3469:
3464:
3460:
3456:
3449:
3445:
3440:
3436:
3431:
3428:
3425:
3421:
3417:
3410:
3407:
3404:
3400:
3395:
3381:
3380:
3379:
3374:
3367:
3360:
3353:
3346:
3339:
3332:
3328:
3324:
3314:
3310:
3306:
3302:
3298:
3294:
3290:
3285:
3281:
3274:
3267:
3260:
3256:
3251:
3247:
3237:
3235:
3231:
3213:
3210:
3206:
3197:
3193:
3175:
3172:
3168:
3147:
3141:
3135:
3132:
3129:
3124:
3121:
3117:
3113:
3108:
3105:
3101:
3097:
3091:
3088:
3082:
3076:
3073:
3070:
3067:
3061:
3058:
3055:
3049:
3035:
3031:
3027:
3010:
3006:
3003:
3000:
2997:
2992:
2988:
2983:
2962:
2959:
2954:
2951:
2948:
2944:
2923:
2920:
2915:
2911:
2902:
2898:
2894:
2876:
2872:
2858:
2849:
2846:
2843:
2839:
2835:
2831:
2826:
2822:
2818:
2815: â
2814:
2810:
2807:
2803:
2797:
2791:
2787:
2781:
2755:
2749:
2746:
2743:
2740:
2737:
2733:
2729:
2726:
2723:
2718:
2715:
2712:
2708:
2704:
2699:
2695:
2690:
2686:
2681:
2677:
2668:
2664:
2643:
2639:
2610:
2606:
2594:
2574:
2571:
2568:
2555:
2552:
2549:
2545:
2541:
2536:
2533:
2530:
2526:
2522:
2519:
2516:
2511:
2508:
2505:
2501:
2497:
2492:
2489:
2486:
2482:
2478:
2473:
2470:
2467:
2463:
2459:
2454:
2451:
2448:
2444:
2440:
2435:
2431:
2427:
2422:
2418:
2409:
2404:
2392:
2388:
2384:
2379:
2375:
2371:
2368:
2365:
2360:
2357:
2354:
2350:
2346:
2341:
2338:
2335:
2331:
2327:
2322:
2319:
2316:
2312:
2308:
2303:
2300:
2297:
2293:
2289:
2284:
2280:
2276:
2271:
2267:
2258:
2242:
2238:
2234:
2217:
2213:
2204:
2200:
2179:
2175:
2171:
2166:
2163:
2160:
2156:
2152:
2149:
2146:
2141:
2137:
2133:
2128:
2125:
2122:
2118:
2114:
2109:
2105:
2101:
2096:
2092:
2082:
2074:
2070:
2066:
2061:
2057:
2053:
2050:
2047:
2042:
2038:
2034:
2029:
2025:
2021:
2016:
2012:
2008:
2003:
1999:
1984:
1981:
1977:
1958:
1955:
1950:
1947:
1944:
1940:
1936:
1933:
1930:
1925:
1921:
1911:
1905:
1902:
1897:
1893:
1889:
1886:
1883:
1878:
1875:
1872:
1868:
1853:
1852:
1846:
1844:
1841:
1840:countable set
1836:
1832:
1813:
1810:
1802:
1798:
1794:
1789:
1785:
1781:
1778:
1775:
1770:
1766:
1762:
1757:
1753:
1738:
1722:
1714:
1710:
1706:
1701:
1697:
1693:
1690:
1687:
1682:
1679:
1676:
1672:
1662:
1654:
1650:
1646:
1641:
1637:
1633:
1630:
1627:
1622:
1618:
1614:
1609:
1605:
1601:
1596:
1592:
1588:
1583:
1579:
1575:
1572:
1569:
1564:
1561:
1558:
1554:
1540:
1539:
1538:
1536:
1529:
1522:
1515:
1512:
1506:
1491:
1477:
1474:
1471:
1468:
1465:
1462:
1457:
1454:
1451:
1447:
1426:
1423:
1420:
1417:
1414:
1411:
1406:
1402:
1380:
1377:
1374:
1371:
1368:
1365:
1360:
1356:
1333:
1329:
1306:
1302:
1281:
1278:
1275:
1272:
1269:
1266:
1261:
1257:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1197:
1194:
1191:
1188:
1185:
1182:
1177:
1173:
1164:
1146:
1142:
1133:
1115:
1111:
1101:
1085:
1081:
1058:
1054:
1031:
1027:
1006:
1000:
995:
991:
968:
964:
943:
937:
932:
928:
918:
916:
892:
889:
886:
881:
877:
853:
850:
845:
841:
814:
810:
790:
786:
782:
779:
775:
771:
766:
762:
759:
758:
756:
746:
744:
743:Eugene Dynkin
740:
736:
732:
728:
723:
719:
717:
713:
708:
704:
700:
696:
692:
691:finite groups
688:
683:
681:
677:
674:, written by
673:
672:Eugene Onegin
668:
664:
660:
655:
654:Andrey Markov
646:
642:
640:
636:
632:
626:
623:
613:
611:
607:
603:
598:
596:
592:
588:
577:
573:
569:
566:
562:
558:
554:
551:
548:
544:
534:
531:
528:
525:The system's
518:
515:
511:
506:
504:
500:
496:
492:
488:
476:
475:Andrey Markov
471:
462:
460:
456:
451:
449:
445:
441:
437:
433:
429:
425:
421:
417:
413:
409:
405:
400:
398:
397:Andrey Markov
394:
390:
386:
382:
378:
374:
370:
366:
363:describing a
362:
358:
354:
342:
337:
335:
330:
328:
323:
322:
320:
319:
314:
311:
309:
306:
305:
304:
303:
298:
295:
293:
290:
288:
285:
283:
280:
278:
275:
273:
270:
269:
268:
267:
262:
259:
257:
254:
252:
249:
247:
244:
243:
242:
241:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
195:
194:
191:
189:
186:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
153:
152:
149:
145:
142:
141:
140:
137:
133:
130:
128:
125:
123:
120:
118:
115:
113:
110:
109:
108:
105:
103:
100:
98:
95:
94:
93:
92:
87:
84:
82:
81:Indeterminism
79:
75:
72:
71:
70:
67:
63:
60:
59:
58:
55:
54:
53:
52:
48:
44:
43:
40:
37:
36:
33:
29:
28:
21:
15905:Graph theory
15786:Econometrics
15748:Wiener space
15636:ItĂŽ integral
15537:Inequalities
15426:Self-similar
15396:GaussâMarkov
15386:Exchangeable
15366:CĂ dlĂ g paths
15302:Risk process
15254:LIBOR market
15123:Random graph
15118:Random field
14930:Superprocess
14868:LĂ©vy process
14863:Jump process
14838:Hunt process
14674:Markov chain
14673:
14554:
14526:
14511:
14504:
14489:
14472:
14448:
14434:
14426:
14414:
14395:
14394:S. P. Meyn.
14384:
14352:
14315:
14290:
14275:
14254:
14238:
14234:
14225:
14218:
14183:
14180:Solar Energy
14179:
14158:. Retrieved
14154:
14141:
14129:. Retrieved
14117:
14113:
14103:
14084:
14080:
14058:
14054:
14020:
14012:
14003:
13999:
13989:
13981:the original
13971:
13960:. Retrieved
13950:
13939:. Retrieved
13935:the original
13924:
13916:
13900:
13892:the original
13882:
13867:
13862:
13843:
13837:
13823:(2): 19â30.
13820:
13816:
13810:
13755:
13751:
13741:
13730:the original
13725:
13713:
13702:the original
13697:
13684:
13657:
13653:
13647:
13633:(1): 27â58.
13630:
13624:
13614:
13579:
13576:Econometrica
13575:
13569:
13560:
13556:
13549:
13522:
13518:
13512:
13503:
13499:
13493:
13468:
13464:
13458:
13439:
13433:
13398:
13394:
13381:
13361:
13354:
13341:
13321:
13307:
13293:
13250:
13246:
13236:
13220:
13216:
13210:
13185:
13181:
13171:
13146:
13143:Solar Energy
13142:
13136:
13111:
13108:Solar Energy
13107:
13101:
13076:
13073:Solar Energy
13072:
13066:
13039:
13036:Solar Energy
13035:
13025:
13000:
12997:Solar Energy
12996:
12990:
12965:
12962:Solar Energy
12961:
12955:
12912:
12908:
12898:
12853:
12849:
12839:
12806:
12802:
12795:
12770:
12766:
12760:
12735:
12731:
12725:
12672:
12668:
12658:
12632:
12626:
12598:
12591:
12556:
12552:
12528:. Retrieved
12515:
12504:
12463:
12454:
12445:
12432:
12413:
12407:
12388:
12384:
12374:
12354:
12347:
12336:. Retrieved
12334:. 2020-03-22
12331:
12322:
12302:
12295:
12284:. Retrieved
12280:
12245:
12239:
12227:. Retrieved
12211:
12200:. Retrieved
12196:
12189:Peres, Yuval
12183:
12164:
12158:
12147:. Retrieved
12132:
12122:
12103:
12097:
12087:
12054:
12050:
12044:
12033:. Retrieved
12013:
12009:
11999:
11980:
11974:
11963:. Retrieved
11951:
11924:
11888:
11879:
11859:
11852:
11842:
11835:
11815:
11808:
11788:
11781:
11754:
11748:
11728:
11698:
11671:
11667:
11660:
11640:
11633:
11608:
11604:
11598:
11578:
11571:
11551:
11544:
11525:
11521:
11491:
11487:
11448:
11438:
11413:
11409:
11403:
11378:
11374:
11365:
11348:
11344:
11338:
11301:
11295:
11275:
11251:(2): 92â96.
11248:
11244:
11213:
11177:
11153:
11148:
11133:
11128:
11112:
11107:
11091:
11086:
11071:
11066:
11046:
11039:
11019:
11012:
10992:
10985:
10965:
10958:
10938:
10901:
10895:
10872:
10861:
10841:
10834:
10814:
10807:
10787:
10662:
10638:
10617:
10608:
10605:
10596:
10592:
10590:
10324:
10308:
10287:middle class
10260:
10249:
10242:
10211:
10191:
10182:
9974:
9901:
9897:
9893:
9889:
9881:
9877:
9873:
9867:
9856:
9834:
9823:
9783:
9777:
9762:
9746:
9734:
9712:Neurobiology
9674:
9665:superlattice
9662:
9650:
9642:
9635:
9630:
9625:
9622:
9483:
9472:
9464:
9461:Applications
9418:
9414:
9411:finite graph
9404:
9385:
9379:
9367:
9310:
9307:Markov model
9301:Markov model
9289:
9285:
9281:
9277:
9273:
9271:
9258:
9254:
9252:
9047:
8989:
8984:
8972:
8964:
8962:
8887:
8885:
8748:
8744:
8735:
8731:
8727:
8723:jump process
8721:
8717:
8713:
8699:
8685:
8683:
8618:
8614:
8612:
8461:
8457:
8417:
8413:
8409:
8407:
8397:hitting time
8396:
8394:
8373:
8368:
8364:
8362:
8250:
8246:
8242:
8238:
8236:
8226:
8222:
8220:
8102:
8097:
8095:
7885:
7386:
7274:
7270:
7268:
7258:
7207:
7205:
7194:
7186:
7181:
7177:
7173:
7171:
7121:
7116:
7112:
7108:
7106:
7041:
7038:
7028:
7024:
7022:
7016:
6985:
6981:
6979:
6866:hitting time
6865:
6861:
6857:
6853:
6849:
6845:
6841:
6837:
6835:
6830:
6803:; otherwise
6774:
6772:
6669:
6663:
6653:
6647:
6641:
6638:
6633:
6629:
6624:
6622:
6612:
6598:
6590:
6583:
6560:
6553:
6546:
6470:
6463:
6459:
6453:
6446:
6440:
6434:
6428:
6421:
6415:
6413:
5591:
5587:
5583:
5579:
5572:
5566:
5560:
5554:
5550:
5546:
5544:
5401:
5397:
5393:
5392:be a length
5389:
5384:
5379:
5374:
5370:
5366:
5362:
5357:
5353:
5347:
5342:
5340:
5213:
5152:
5148:
5141:
5134:
5127:
5123:
5119:
5115:
5111:
5107:
5105:
5100:
5088:
5084:
5080:
5076:
5067:
5027:
5017:
5013:
5009:
5005:
5001:
4996:
4992:
4988:
4984:
4982:
4862:
4858:
4854:
4852:
4847:
4843:
4839:
4835:
4831:
4827:
4823:
4819:
4811:
4807:
4803:
4794:
4790:
4786:
4782:
4773:
4769:
4767:
4687:
4686:Subtracting
4685:
4643:
4583:
4579:
4575:
4572:
4569:
4273:, the limit
4270:
4268:
4214:
4212:
4144:
4138:
4133:
4129:
4125:
4121:
4119:
4052:
4021:
4011:
4007:
3962:
3898:
3856:
3850:
3836:
3832:
3830:
3736:
3728:
3724:
3714:
3702:
3633:
3629:
3627:
3372:
3365:
3358:
3351:
3344:
3337:
3330:
3326:
3322:
3320:
3308:
3304:
3300:
3296:
3292:
3287:follows the
3283:
3279:
3272:
3265:
3258:
3254:
3249:
3245:
3243:
3233:
3229:
3194:, using the
3033:
3032:and for all
3029:
3028:â 0 for all
3025:
2900:
2896:
2892:
2863:
2847:
2841:
2837:
2833:
2829:
2824:
2820:
2816:
2812:
2808:
2801:
2795:
2789:
2785:
2783:
2666:
2662:
2592:
2240:
2236:
2202:
2198:
1979:
1975:
1842:
1834:
1830:
1828:
1527:
1520:
1513:
1508:
1162:
1131:
1102:
919:
914:
833:draws, with
800:
761:Random walks
720:
684:
652:
643:
627:
619:
599:
595:Markov model
590:
586:
583:
557:Harris chain
524:
507:
484:
458:
454:
452:
401:
372:
356:
353:Markov chain
352:
350:
313:Tree diagram
308:Venn diagram
272:Independence
218:Markov chain
217:
102:Sample space
15831:Ruin theory
15769:Disciplines
15641:ItĂŽ's lemma
15416:Predictable
15091:Percolation
15074:Potts model
15069:Ising model
15033:White noise
14991:Differences
14853:ItĂŽ process
14793:Cox process
14689:Loop-erased
14684:Random walk
14471:Seneta, E.
14255:Probability
14186:: 688â695.
14120:(1): 7â14.
14087:: 152â158.
13149:: 174â183.
13079:: 487â495.
13042:: 229â242.
13003:: 160â170.
12507:, Springer
12010:SIAM Review
11102:(Table 6.1)
10868:"Markovian"
10273:Das Kapital
9870:M/M/1 queue
9752:solar power
9486:lattice QCD
9267:unit vector
8886:From this,
8747:into state
8613:For a CTMC
8379:time series
8223:irreducible
8098:irreducible
7191:Terminology
6864:, the mean
6673:. That is:
6645:has period
6634:irreducible
6625:communicate
5388:. Also let
5156:). Then by
5122:, that is,
4800:zero matrix
4106:dot product
3901:eigenvector
2665:-tuples of
1132:total value
785:number line
625:terminate.
616:Transitions
602:state space
527:state space
510:state space
503:independent
499:conditional
369:probability
228:Random walk
69:Determinism
57:Probability
15889:Categories
15841:Statistics
15621:Filtration
15522:Kolmogorov
15506:Blumenthal
15431:Stationary
15371:Continuous
15359:Properties
15244:HullâWhite
14986:Martingale
14873:Local time
14761:Fractional
14739:pure birth
14324:. Berlin:
14301:. online:
14272:J. L. Doob
14211:References
13962:2009-04-24
13941:2007-11-26
13876:1576470792
12530:2017-06-02
12448:: 781â784.
12338:2024-02-01
12286:2024-02-01
12281:bactra.org
12202:2024-02-01
12149:2016-03-04
12035:2021-05-31
11965:2017-06-02
11371:Seneta, E.
11132:Dodge, Y.
10911:3540047581
10889:required.)
10283:capitalism
10188:Statistics
9692:nucleotide
9653:copolymers
9288:(ÎŽ),
9284:(0),
9010:such that
8686:reversible
8127:such that
7479:goes like
7103:Ergodicity
7027:is called
6858:persistent
6619:Properties
4814:must be a
4016:eigenvalue
1849:Variations
610:Variations
481:Definition
465:Principles
383:(DTMC). A
139:Experiment
86:Randomness
32:statistics
15753:Classical
14766:Geometric
14756:Excursion
14561:EMS Press
14389:Fizmatgiz
14131:5 October
13768:CiteSeerX
13662:CiteSeerX
13660:: 49â83.
13584:CiteSeerX
13525:: 21â86.
13506:: 425â40.
13444:CiteSeerX
13411:CiteSeerX
13223:: 81â91,
13202:1134-3060
12682:1209.6210
11985:CiteSeerX
11674:(1): 57.
11494:(1): 33.
11306:CiteSeerX
10629:AstroTurf
10291:political
10267:Karl Marx
10247:setting.
10236:model of
10168:α
10142:α
10139:−
10107:α
10104:−
10080:α
9953:α
9950:−
9926:α
9770:systems.
9645:in silico
9588:Catalytic
9580:⟶
9565:Substrate
9551:⇀
9544:−
9533:−
9526:↽
9492:Chemistry
9217:−
9200:
9191:φ
9176:−
9159:
9150:φ
9147:−
9138:π
9104:‖
9100:φ
9097:‖
9076:φ
9056:φ
9030:φ
9021:φ
8998:φ
8975:) is the
8971:and diag(
8940:−
8922:
8910:−
8860:otherwise
8844:≠
8810:≠
8803:∑
8660:−
8637:^
8583:∉
8531:∈
8524:∑
8520:−
8510:∈
8304:∈
8206:Ω
8186:∅
8140:−
8093:instead.
8073:Σ
8066:Ω
8043:…
8018:…
7966:Ω
7963:→
7960:Ω
7933:Σ
7905:Σ
7898:Ω
7835:−
7799:≤
7749:−
7731:≤
7705:−
7696:≤
7603:−
7597:−
7588:≤
7565:≥
7508:→
7502:→
7499:⋯
7496:→
7490:→
7428:−
7419:≤
7208:primitive
7051:π
7029:absorbing
6938:⋅
6930:∞
6915:∑
6854:recurrent
6842:transient
6831:aperiodic
6730:∣
6519:λ
6510:≥
6507:⋯
6504:≥
6490:λ
6353:λ
6343:λ
6319:⋯
6276:λ
6266:λ
6205:λ
6195:λ
6126:λ
6109:≠
6088:⊥
6042:λ
6025:⋯
5989:λ
5942:λ
5913:−
5896:Σ
5851:⋯
5764:−
5747:Σ
5717:−
5709:Σ
5697:⋯
5685:−
5677:Σ
5656:−
5648:Σ
5611:π
5522:∈
5466:∑
5312:λ
5303:≥
5300:⋯
5297:≥
5283:λ
5274:≥
5260:λ
5237:λ
5191:−
5183:Σ
5045:π
5037:π
4948:−
4925:−
4714:−
4613:∞
4610:→
4292:∞
4289:→
4240:∞
4237:→
4198:π
4173:∞
4170:→
4080:π
4076:⋅
4064:∑
4032:π
3982:π
3972:∑
3927:∑
3914:π
3881:π
3870:π
3794:∣
3739:equal to
3600:−
3506:…
3437:∣
3236:happens.
3169:δ
3102:δ
3074:∣
3024:, and as
2832:to state
2741:−
2727:…
2716:−
2553:−
2534:−
2520:…
2509:−
2490:−
2471:−
2452:−
2441:∣
2369:…
2358:−
2339:−
2320:−
2301:−
2290:∣
2150:…
2051:…
1948:−
1937:∣
1890:∣
1779:…
1694:∣
1631:…
1576:∣
1466:ℓ
1455:−
1227:×
1221:×
1004:$
1001:≥
941:$
893:∈
455:Markovian
428:economics
424:chemistry
132:Singleton
15874:Category
15758:Abstract
15292:BĂŒhlmann
14898:Compound
14571:Archived
14312:(1965).
14160:26 March
13909:Archived
13802:22198760
13330:Archived
13285:25984837
12947:27429455
12890:19816557
12831:22186291
12787:19527020
12752:19527020
12717:23408514
12583:26968853
12219:(2016).
12079:22181092
11445:Heyde CC
10920:52203046
10675:See also
10614:Baseball
10427:♭
10408:♯
10384:♭
10376:♯
10333:such as
10331:software
10277:, tying
9977:PageRank
9801:through
9226:‖
9187:‖
8837:if
8708:, of an
8172:implies
7925:, where
7853:, where
7275:exponent
7107:A state
7023:A state
6836:A state
6775:periodic
6639:A state
5114:and let
4802:of size
4781:of size
4014:with an
3660:′
2899:at time
2239:) where
2197:for all
1974:for all
1735:if both
749:Examples
685:In 1912
631:integers
365:sequence
213:Variance
15381:Ergodic
15269:VaĆĄĂÄek
15111:Poisson
14771:Meander
14586:YouTube
14563:, 2001
14447:(1960)
14274:(1953)
14188:Bibcode
13793:3271566
13760:Bibcode
13606:1912559
13485:2227127
13403:Bibcode
13276:4434998
13255:Bibcode
13151:Bibcode
13116:Bibcode
13081:Bibcode
13044:Bibcode
13005:Bibcode
12970:Bibcode
12938:4946376
12917:Bibcode
12881:2749218
12858:Bibcode
12811:Bibcode
12708:3568780
12687:Bibcode
12574:5862921
12229:22 June
12059:Bibcode
12030:2132659
11613:Bibcode
11430:1403518
11395:1403785
11136:, OUP.
11115:, OUP.
11074:. CUP.
10621:bunting
10601:phrasal
9794:entropy
9731:Testing
9671:Biology
9568:binding
9469:Physics
9423:measure
8967:is the
8710:ergodic
8227:ergodic
7873:is the
7787:is the
7259:regular
7113:ergodic
6657:is the
5361:be the
5126:= diag(
4798:is the
4777:is the
4110:simplex
3733:element
3190:is the
1838:form a
1490:state.
649:History
440:physics
432:finance
420:biology
393:Russian
127:Outcome
15721:Tanaka
15406:Mixing
15401:Markov
15274:Wilkie
15239:HoâLee
15234:Heston
15006:Super-
14751:Bridge
14699:Biased
14533:
14518:
14496:
14479:
14465:
14455:
14443:&
14402:
14369:
14340:
14297:
14282:
14264:
14029:
13874:
13850:
13800:
13790:
13770:
13664:
13604:
13586:
13483:
13446:
13413:
13369:
13283:
13273:
13200:
12945:
12935:
12888:
12878:
12829:
12785:
12750:
12715:
12705:
12647:
12614:
12581:
12571:
12478:
12420:
12362:
12310:
12252:
12171:
12140:
12077:
12028:
11987:
11939:
11903:
11867:
11823:
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11769:
11736:
11706:
11648:
11586:
11559:
11463:
11428:
11393:
11326:
11308:
11283:
11221:
11189:
11185:â466.
11140:
11119:
11098:
11078:
11054:
11027:
11000:
10973:
10946:
10918:
10908:
10849:
10822:
10795:
10667:, and
10341:, and
10335:Csound
9981:Google
9696:genome
9441:, the
9437:, the
8963:where
8096:Since
7767:where
6980:State
6630:closed
6545:hence
6414:Since
5341:Since
4785:, and
4768:where
4578:be an
4213:where
3721:matrix
3717:finite
3628:where
3198:. The
3160:where
459:Markov
446:, and
74:System
62:Axioms
15574:Tools
15350:M/M/c
15345:M/M/1
15340:M/G/1
15330:Fluid
14996:Local
14387:) by
14151:(PDF)
13733:(PDF)
13722:(PDF)
13705:(PDF)
13694:(PDF)
13602:JSTOR
13481:JSTOR
13391:(PDF)
12677:arXiv
12525:(PDF)
12442:(PDF)
12224:(PDF)
12026:JSTOR
11960:(PDF)
11426:JSTOR
11391:JSTOR
10883:
10769:Notes
10571:0.25
10529:0.75
10473:0.22
10450:Notes
10321:Music
10313:and "
10305:Games
9417:or a
9409:of a
9048:with
8677:. By
7273:, or
5549:with
5405:span
5147:,...,
4979:ones.
3839:is a
2628:from
1163:count
375:." A
359:is a
107:Event
15526:LĂ©vy
15325:Bulk
15209:Chen
15001:Sub-
14959:Both
14531:ISBN
14516:ISBN
14494:ISBN
14477:ISBN
14463:ISBN
14453:ISBN
14408:CTCN
14400:ISBN
14367:ISBN
14338:ISBN
14303:MCSS
14295:ISBN
14280:ISBN
14262:ISBN
14162:2024
14133:2023
14027:ISBN
14000:BYTE
13872:ISBN
13848:ISBN
13798:PMID
13553:e.g.
13367:ISBN
13281:PMID
13198:ISSN
12943:PMID
12886:PMID
12827:PMID
12783:PMID
12748:PMID
12713:PMID
12645:ISBN
12612:ISBN
12579:PMID
12476:ISBN
12418:ISBN
12360:ISBN
12308:ISBN
12250:ISBN
12231:2024
12169:ISBN
12138:ISBN
12075:PMID
11937:ISBN
11901:ISBN
11865:ISBN
11821:ISBN
11794:ISBN
11767:ISBN
11734:ISBN
11704:ISBN
11646:ISBN
11584:ISBN
11557:ISBN
11461:ISBN
11324:ISBN
11281:ISBN
11219:ISBN
11187:ISBN
11138:ISBN
11117:ISBN
11096:ISBN
11076:ISBN
11052:ISBN
11025:ISBN
10998:ISBN
10971:ISBN
10944:ISBN
10916:OCLC
10906:ISBN
10847:ISBN
10820:ISBN
10793:ISBN
10623:and
10568:0.25
10557:0.2
10523:0.25
10501:0.1
10498:0.75
10495:0.15
10467:0.18
10419:0.7
10416:0.05
10413:0.25
10400:0.3
10367:Note
10352:MIDI
10043:has
9975:The
9837:LZMA
9835:The
9813:and
9682:and
9591:step
9477:and
9453:and
9197:diag
9156:diag
8919:diag
8395:The
7554:has
7269:The
6856:(or
6788:>
6752:>
6699:>
6615:or.
6607:and
5251:>
5106:Let
5087:has
3731:)th
3004:<
2864:Let
2804:, a
2572:>
2201:and
1811:>
1007:0.60
944:0.50
792:one.
705:and
697:and
695:Paul
629:the
570:Any
457:and
15106:Cox
14584:on
14359:doi
14330:doi
14243:doi
14196:doi
14184:184
14122:doi
14089:doi
14085:122
13825:doi
13788:PMC
13778:doi
13756:108
13672:doi
13635:doi
13631:105
13594:doi
13535:hdl
13527:doi
13473:doi
13421:doi
13271:PMC
13263:doi
13225:doi
13190:doi
13159:doi
13147:170
13124:doi
13089:doi
13077:173
13052:doi
13040:115
13013:doi
13001:103
12978:doi
12933:PMC
12925:doi
12876:PMC
12866:doi
12819:doi
12775:doi
12740:doi
12703:PMC
12695:doi
12637:doi
12604:doi
12569:PMC
12561:doi
12503:",
12468:doi
12393:doi
12389:158
12108:doi
12067:doi
12018:doi
11929:doi
11893:doi
11759:doi
11676:doi
11621:doi
11530:doi
11496:doi
11453:doi
11418:doi
11383:doi
11353:doi
11316:doi
11253:doi
11249:101
11183:464
10593:and
10576:GD
10565:0.5
10562:GA
10554:0.4
10551:0.4
10548:GG
10540:0.1
10537:0.9
10534:DG
10520:DA
10506:DD
10492:AG
10484:0.5
10481:0.5
10478:AD
10470:0.6
10464:AA
10435:0.3
10432:0.7
10397:0.6
10394:0.1
10339:Max
10297:to
10269:'s
9892:to
9876:to
9861:).
9433:of
9386:any
9269:.)
8363:If
8198:or
8103:In
7621:If
7534:If
6988:if
6984:is
6777:if
6690:gcd
6651:if
6458:as
6452:=
5590:as
5582:...
5580:xPP
5099:of
4832:n+1
4603:lim
4282:lim
4230:lim
4163:lim
3735:of
3641:(a
3232:to
1236:216
915:not
913:is
633:or
373:now
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2455:1
2449:n
2445:X
2436:n
2432:x
2428:=
2423:n
2419:X
2415:(
2405:=
2398:)
2393:1
2389:x
2385:=
2380:1
2376:X
2372:,
2366:,
2361:2
2355:n
2351:x
2347:=
2342:2
2336:n
2332:X
2328:,
2323:1
2317:n
2313:x
2309:=
2304:1
2298:n
2294:X
2285:n
2281:x
2277:=
2272:n
2268:X
2264:(
2241:m
2237:m
2218:0
2214:X
2203:k
2199:n
2185:)
2180:k
2176:x
2172:=
2167:k
2164:+
2161:n
2157:X
2153:,
2147:,
2142:1
2138:x
2134:=
2129:1
2126:+
2123:n
2119:X
2115:,
2110:0
2106:x
2102:=
2097:n
2093:X
2089:(
2083:=
2080:)
2075:k
2071:x
2067:=
2062:k
2058:X
2054:,
2048:,
2043:1
2039:x
2035:=
2030:1
2026:X
2022:,
2017:0
2013:x
2009:=
2004:0
2000:X
1996:(
1982:.
1980:n
1976:n
1962:)
1959:y
1956:=
1951:1
1945:n
1941:X
1934:x
1931:=
1926:n
1922:X
1918:(
1912:=
1909:)
1906:y
1903:=
1898:n
1894:X
1887:x
1884:=
1879:1
1876:+
1873:n
1869:X
1865:(
1843:S
1835:i
1831:X
1808:)
1803:n
1799:x
1795:=
1790:n
1786:X
1782:,
1776:,
1771:1
1767:x
1763:=
1758:1
1754:X
1750:(
1723:,
1720:)
1715:n
1711:x
1707:=
1702:n
1698:X
1691:x
1688:=
1683:1
1680:+
1677:n
1673:X
1669:(
1663:=
1660:)
1655:n
1651:x
1647:=
1642:n
1638:X
1634:,
1628:,
1623:2
1619:x
1615:=
1610:2
1606:X
1602:,
1597:1
1593:x
1589:=
1584:1
1580:X
1573:x
1570:=
1565:1
1562:+
1559:n
1555:X
1551:(
1531:3
1528:X
1524:2
1521:X
1517:1
1514:X
1478:p
1475:,
1472:m
1469:,
1463:=
1458:1
1452:n
1448:X
1427:k
1424:,
1421:j
1418:,
1415:i
1412:=
1407:n
1403:X
1381:1
1378:,
1375:0
1372:,
1369:1
1366:=
1361:2
1357:X
1334:1
1330:X
1307:2
1303:X
1282:0
1279:,
1276:1
1273:,
1270:0
1267:=
1262:1
1258:X
1233:=
1230:6
1224:6
1218:6
1198:5
1195:,
1192:0
1189:,
1186:1
1183:=
1178:6
1174:X
1147:n
1143:X
1116:n
1112:X
1086:6
1082:X
1059:7
1055:X
1032:6
1028:X
996:7
992:X
969:6
965:X
938:=
933:6
929:X
901:}
897:N
890:n
887::
882:n
878:X
874:{
854:0
851:=
846:0
842:X
831:n
815:n
811:X
340:e
333:t
326:v
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