1865:
8632:
1580:
8473:
5482:
10678:
10082:
9774:
9521:
1860:{\displaystyle {\begin{aligned}{}&X_{n}\ \xrightarrow {d} \ X,\ \ X_{n}\ \xrightarrow {\mathcal {D}} \ X,\ \ X_{n}\ \xrightarrow {\mathcal {L}} \ X,\ \ X_{n}\ \xrightarrow {d} \ {\mathcal {L}}_{X},\\&X_{n}\rightsquigarrow X,\ \ X_{n}\Rightarrow X,\ \ {\mathcal {L}}(X_{n})\to {\mathcal {L}}(X),\\\end{aligned}}}
8627:{\displaystyle {\begin{matrix}{\xrightarrow {\overset {}{L^{s}}}}&{\underset {s>r\geq 1}{\Rightarrow }}&{\xrightarrow {\overset {}{L^{r}}}}&&\\&&\Downarrow &&\\{\xrightarrow {\text{a.s.}}}&\Rightarrow &{\xrightarrow {p}}&\Rightarrow &{\xrightarrow {d}}\end{matrix}}}
5749:
62:
and they formalize the idea that certain properties of a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence
116:
While the above discussion has related to the convergence of a single series to a limiting value, the notion of the convergence of two series towards each other is also important, but this is easily handled by studying the sequence defined as either the difference or the ratio of the two series.
3681:
represent the distribution of possible outputs by the algorithm. Because the pseudorandom number is generated deterministically, its next value is not truly random. Suppose that as you observe a sequence of randomly generated numbers, you can deduce a pattern and make increasingly accurate
5337:
845:
Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is very frequently used in practice; most often it arises from application of the
4115:
4531:
9177:
7251:
9060:
9313:
8953:
5774:
on the space of random variables. This means there is no topology on the space of random variables such that the almost surely convergent sequences are exactly the converging sequences with respect to that topology. In particular, there is no metric of almost sure
10536:
8857:
8728:
63:
relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution.
5326:
9920:
9612:
5625:
9367:
6030:
4246:
785:
5909:
502:
Suppose a new dice factory has just been built. The first few dice come out quite biased, due to imperfections in the production process. The outcome from tossing any of them will follow a distribution markedly different from the desired
3852:
4937:
10344:
6038:
by using sure convergence compared to using almost sure convergence. The difference between the two only exists on sets with probability zero. This is why the concept of sure convergence of random variables is very rarely used.
4631:
Not every sequence of random variables which converges to another random variable in distribution also converges in probability to that random variable. As an example, consider a sequence of standard normal random variables
2161:
8173:
6829:
4415:
5144:
3088:
2990:
828:
663:
5558:
8450:
2063:
39:. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit
8242:
7917:
8091:
8040:
7559:
7508:
2895:
3989:
2581:
3379:
3289:
11620:
6964:
3180:
1986:
10964:
4994:
Consider a man who tosses seven coins every morning. Each afternoon, he donates one pound to a charity for each head that appeared. The first time the result is all tails, however, he will stop permanently.
2333:
10867:
8361:
8303:
7986:
7704:
7646:
265:
9075:
7303:
6299:
7835:
7784:
7425:
7374:
10807:
5477:{\displaystyle \mathbb {P} {\Bigl (}\limsup _{n\to \infty }{\bigl \{}\omega \in \Omega :|X_{n}(\omega )-X(\omega )|>\varepsilon {\bigr \}}{\Bigr )}=0\quad {\text{for all}}\quad \varepsilon >0.}
5227:
2803:
2687:
7106:
2499:
provides several equivalent definitions of convergence in distribution. Although these definitions are less intuitive, they are used to prove a number of statistical theorems. The lemma states that
10744:
8968:
5218:
481:
6419:
10673:{\displaystyle \left.{\begin{matrix}X_{n}\xrightarrow {\overset {}{\text{a.s.}}} X\\|X_{n}|<Y\\\mathbb {E} <\infty \end{matrix}}\right\}\quad \Rightarrow \quad X_{n}\xrightarrow {L^{1}} X}
1585:
9225:
1051:
8868:
8772:
71:"Stochastic convergence" formalizes the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle into a pattern. The pattern may for instance be
5937:
1914:
8646:
2624:
1203:
6526:
43:
of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution.
10494:
6194:
4391:
3966:
are independent (and thus convergence in probability is a condition on the joint cdf's, as opposed to convergence in distribution, which is a condition on the individual cdf's), unless
2423:
6133:
4621:
3974:
cannot, whenever the deterministic value is a discontinuity point (not isolated), be handled by convergence in distribution, where discontinuity points have to be explicitly excluded.
2199:
6234:
5826:
10077:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ X,\ \ Y_{n}\ {\xrightarrow {\overset {}{p}}}\ Y\ \quad \Rightarrow \quad (X_{n},Y_{n})\ {\xrightarrow {\overset {}{p}}}\ (X,Y)}
9769:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{d}}}\ X,\ \ Y_{n}\ {\xrightarrow {\overset {}{d}}}\ c\ \quad \Rightarrow \quad (X_{n},Y_{n})\ {\xrightarrow {\overset {}{d}}}\ (X,c)}
4332:
1398:
179:
9516:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{d}}}\ X,\ \ |X_{n}-Y_{n}|\ {\xrightarrow {\overset {}{p}}}\ 0\ \quad \Rightarrow \quad Y_{n}\ {\xrightarrow {\overset {}{d}}}\ X}
4716:
955:
902:
4795:
1082:
510:
As the factory is improved, the dice become less and less loaded, and the outcomes from tossing a newly produced die will follow the uniform distribution more and more closely.
11016:
4980:
Consider an animal of some short-lived species. We record the amount of food that this animal consumes per day. This sequence of numbers will be unpredictable, but we may be
4163:
6721:
6692:
2736:
11845:
6458:
838:
Loosely, with this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given
6352:
1501:
1365:
1275:
6559:
6078:
711:
8765:
5744:{\displaystyle \mathbb {P} {\Bigl (}\omega \in \Omega \colon \,d{\big (}X_{n}(\omega ),X(\omega ){\big )}\,{\underset {n\to \infty }{\longrightarrow }}\,0{\Bigr )}=1}
1323:
414:
1459:
1424:
9850:
9603:
6871:
5620:
4770:
4743:
4657:
1156:
370:
315:
343:
7730:
7585:
7451:
3767:
1547:
1233:
5077:
6891:
6619:
6599:
6579:
6319:
4959:
4790:
4292:
3403:
3313:
3223:
3203:
3111:
3013:
2915:
2826:
2714:
1567:
1521:
1297:
1129:
1102:
434:
288:
10266:
11624:
12052:
3721:
The basic idea behind this type of convergence is that the probability of an âunusualâ outcome becomes smaller and smaller as the sequence progresses.
2083:
12069:
11289:
8096:
6744:
5331:
842:. More precisely, the distribution of the associated random variable in the sequence becomes arbitrarily close to a specified fixed distribution.
8467:
The chain of implications between the various notions of convergence are noted in their respective sections. They are, using the arrow notation:
3020:
2922:
602:
5498:
8366:
1997:
3728:
if it converges in probability to the quantity being estimated. Convergence in probability is also the type of convergence established by the
11489:
8186:
7840:
4110:{\displaystyle X_{n}\ \xrightarrow {p} \ X,\ \ X_{n}\ \xrightarrow {P} \ X,\ \ {\underset {n\to \infty }{\operatorname {plim} }}\,X_{n}=X.}
3567:
8045:
7994:
7513:
7462:
2833:
2519:
3320:
3230:
11056:
is essentially a question of convergence, and many of the same concepts and relationships used above apply to the continuity question.
6908:
4526:{\displaystyle d(X,Y)=\inf \!{\big \{}\varepsilon >0:\ \mathbb {P} {\big (}|X-Y|>\varepsilon {\big )}\leq \varepsilon {\big \}}}
3118:
1927:
11034:
10874:
2269:
113:
These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied.
11869:
10814:
9172:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{L^{s}}}}\ X,}
8308:
8250:
7930:
7651:
7593:
695:
195:
7259:
7246:{\displaystyle {\overset {}{X_{n}\xrightarrow {L^{r}} X}}\quad \Rightarrow \quad \lim _{n\to \infty }\mathbb {E} =\mathbb {E} .}
6239:
7789:
7738:
7379:
7328:
4536:
10763:
9055:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{p}}}\ X}
2747:
2631:
11578:
11559:
11540:
11518:
11499:
11470:
11447:
11428:
11409:
11382:
11363:
11327:
11223:
11069:
3606:
4260:
In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable
10704:
9308:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{d}}}\ c\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{p}}}\ c,}
5179:
3677:
Suppose that a random number generator generates a pseudorandom floating point number between 0 and 1. Let random variable
442:
8948:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ X\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{d}}}\ X}
6360:
12001:
11652:
8852:{\displaystyle X_{n}\ \xrightarrow {\overset {}{p}} \ X\quad \Rightarrow \quad X_{n_{k}}\ \xrightarrow {\text{a.s.}} \ X}
6034:
Sure convergence of a random variable implies all the other kinds of convergence stated above, but there is no payoff in
986:
12162:
11927:
8723:{\displaystyle X_{n}\ {\xrightarrow {\text{a.s.}}}\ X\quad \Rightarrow \quad X_{n}\ {\xrightarrow {\overset {}{p}}}\ X}
12064:
11983:
2220:, and even to the ârandom variablesâ which are not measurable â a situation which occurs for example in the study of
1886:
1159:
699:
124:
90:
That the probability distribution describing the next outcome may grow increasingly similar to a certain distribution
9070:-th order mean implies convergence in lower order mean, assuming that both orders are greater than or equal to one:
2586:
6461:
1165:
974:
909:
6467:
3693:
random numbers. As you learn the pattern and your guesses become more accurate, not only will the distribution of
3548:
10439:
6138:
5321:{\displaystyle \mathbb {P} {\Bigl (}\omega \in \Omega :\lim _{n\to \infty }X_{n}(\omega )=X(\omega ){\Bigr )}=1.}
504:
11616:
10188:. Usually, convergence in distribution does not imply convergence almost surely. However, for a given sequence {
3724:
The concept of convergence in probability is used very often in statistics. For example, an estimator is called
2376:
11963:
11943:
6083:
4338:
2169:
6199:
12157:
12059:
11993:
11897:
11780:
11064:
11049:
10520:
11239:
2487:
2448:
2232:
3599:
1370:
825:
grows larger, the shape of the probability density function gets closer and closer to the
Gaussian curve.
130:
12047:
11917:
4662:
4297:
914:
861:
6025:{\displaystyle \left\{\omega \in \Omega :\lim _{n\to \infty }X_{n}(\omega )=X(\omega )\right\}=\Omega .}
12011:
11953:
11810:
11084:
8637:
These properties, together with a number of other special cases, are summarized in the following list:
4271:
4241:{\displaystyle \forall \varepsilon >0,\mathbb {P} {\big (}d(X_{n},X)\geq \varepsilon {\big )}\to 0.}
3970:
is deterministic like for the weak law of large numbers. At the same time, the case of a deterministic
3411:
1277:
1059:
5763:), and hence implies convergence in distribution. It is the notion of convergence used in the strong
12016:
11948:
10971:
3729:
373:
11968:
12042:
12021:
11958:
10413:
839:
827:
40:
6704:
6675:
2719:
2212:
The definition of convergence in distribution may be extended from random vectors to more general
780:{\displaystyle \scriptstyle Z_{n}={\scriptscriptstyle {\frac {1}{\sqrt {n}}}}\sum _{i=1}^{n}X_{i}}
11823:
11645:
11059:
11039:
6424:
5904:{\displaystyle \forall \omega \in \Omega \colon \ \lim _{n\to \infty }X_{n}(\omega )=X(\omega ),}
1206:
6324:
1464:
1328:
1238:
586:
grows larger, this distribution will gradually start to take shape more and more similar to the
11973:
11902:
11795:
11775:
11044:
7078:
6531:
6050:
4143:
541:
11355:
11800:
11692:
11074:
8737:
5925:
5032:
3580:
3382:
3292:
2224:. This is the âweak convergence of laws without laws being definedâ â except asymptotically.
1302:
847:
788:
670:
576:
386:
377:
7306:
3847:{\displaystyle \lim _{n\to \infty }\mathbb {P} {\big (}|X_{n}-X|>\varepsilon {\big )}=0.}
1429:
1403:
11978:
11864:
11749:
11480:
11079:
11019:
10754:
9823:
9576:
6849:
6670:
5764:
5593:
4932:{\displaystyle P(|X_{n}-Y_{n}|\geq \epsilon )=P(|X_{n}|\cdot |(1-(-1)^{n})|\geq \epsilon )}
4748:
4721:
4635:
3725:
3653:
1134:
348:
293:
2447:
In general, convergence in distribution does not imply that the sequence of corresponding
328:
81:
An increasing similarity of outcomes to what a purely deterministic function would produce
8:
12134:
11922:
11805:
11744:
11713:
11598:
7709:
7564:
7430:
2694:
1526:
1212:
1105:
587:
75:
10394:
sufficiently quickly (i.e. the above sequence of tail probabilities is summable for all
12104:
12006:
11912:
11859:
11785:
11718:
11638:
11348:
11233:
11053:
10339:{\displaystyle \sum _{n}\mathbb {P} \left(|X_{n}-X|>\varepsilon \right)<\infty ,}
6876:
6604:
6584:
6564:
6304:
6035:
4944:
4775:
4277:
3388:
3298:
3208:
3188:
3096:
2998:
2900:
2811:
2806:
2699:
2441:
1552:
1506:
1282:
1114:
1087:
419:
273:
51:
20:
3981:
over an arrow indicating convergence, or using the "plim" probability limit operator:
11892:
11574:
11555:
11536:
11528:
11514:
11495:
11466:
11443:
11424:
11405:
11378:
11359:
11323:
11219:
5918:
5798:
2496:
2221:
437:
11290:"real analysis - Generalizing Scheffe's Lemma using only Convergence in Probability"
376:. Other forms of convergence are important in other useful theorems, including the
12094:
12033:
11854:
11754:
11709:
11697:
11458:
11397:
7318:
5929:
5785:
5760:
3649:
2690:
2451:
will also converge. As an example one may consider random variables with densities
2432:
to be in a given range is approximately equal to the probability that the value of
78:
in the classical sense to a fixed value, perhaps itself coming from a random event
16:
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
9792:
converges to a constant is important, if it were to converge to a random variable
5025:
of days, there is a nonzero probability the terminating condition will not occur.
4401:
on the space of random variables over a fixed probability space. This topology is
3629:
Consider the following experiment. First, pick a random person in the street. Let
11476:
2156:{\displaystyle \lim _{n\to \infty }\mathbb {P} (X_{n}\in A)=\mathbb {P} (X\in A)}
905:
106:
11341:. Wiley Series in Probability and Mathematical Statistics (2nd ed.). Wiley.
11318:
Bickel, Peter J.; Klaassen, Chris A.J.; Ritov, Yaâacov; Wellner, Jon A. (1998).
3949:
Notice that for the condition to be satisfied, it is not possible that for each
2065:
the convergence in distribution is defined similarly. We say that this sequence
11907:
11723:
8168:{\displaystyle aX_{n}+bY_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ aX+bY}
6835:
6581:
almost everywhere (in fact the set on which this sequence does not converge to
5802:
5577:
3689:
be your guess of the value of the next random number after observing the first
3183:
2739:
2213:
2202:
99:
11401:
12151:
12119:
12114:
12099:
12089:
11790:
11704:
11679:
7453:
6824:{\displaystyle \lim _{n\to \infty }\mathbb {E} \left(|X_{n}-X|^{r}\right)=0,}
5172:
5036:
3637:
a random variable. Then ask other people to estimate this height by eye. Let
1992:
11876:
11675:
5914:
5588:
5139:{\displaystyle \mathbb {P} \!\left(\lim _{n\to \infty }\!X_{n}=X\right)=1.}
4984:
that one day the number will become zero, and will stay zero forever after.
3083:{\displaystyle \lim \sup \mathbb {P} (X_{n}\in F)\leq \mathbb {P} (X\in F)}
2985:{\displaystyle \lim \inf \mathbb {P} (X_{n}\in G)\geq \mathbb {P} (X\in G)}
2217:
658:{\displaystyle \scriptstyle Z_{n}={\frac {\sqrt {n}}{\sigma }}(X_{n}-\mu )}
7097:-th mean. Hence, convergence in mean square implies convergence in mean.
5553:{\displaystyle {\overset {}{X_{n}\,{\xrightarrow {\mathrm {a.s.} }}\,X.}}}
11612:
8445:{\displaystyle aX_{n}+bY_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ aX+bY}
2058:{\displaystyle \left\{X_{1},X_{2},\dots \right\}\subset \mathbb {R} ^{k}}
87:
An increasing "aversion" against straying far away from a certain outcome
55:
5018:
that one day this amount will be zero, and stay zero forever after that.
46:
The concept is important in probability theory, and its applications to
12129:
11820:
8237:{\displaystyle X_{n}Y_{n}{\xrightarrow {\overset {}{\text{a.s.}}}}\ XY}
4402:
3682:
predictions as to what the next randomly generated number will be. Let
3595:
3091:
47:
8463:
None of the above statements are true for convergence in distribution.
12124:
12109:
7912:{\displaystyle aX_{n}+bY_{n}\ {\xrightarrow {\overset {}{p}}}\ aX+bY}
10836:
10784:
10723:
10654:
10561:
10046:
9985:
9942:
9738:
9677:
9634:
9497:
9455:
9389:
9286:
9247:
9143:
9097:
9036:
8990:
8929:
8890:
8837:
8793:
8704:
8668:
8613:
8594:
8575:
8536:
8486:
8407:
8330:
8272:
8215:
8137:
8067:
8016:
7959:
7881:
7811:
7760:
7673:
7615:
7535:
7484:
7401:
7350:
7281:
7127:
6931:
5521:
4365:
4315:
4048:
4010:
1947:
1727:
1687:
1647:
1609:
669:
to the standard normal, the result that follows from the celebrated
11764:
11733:
11684:
11661:
8086:{\displaystyle Y_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ Y}
8035:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ X}
7554:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ Y}
7503:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{\text{a.s.}}}}\ X}
6654:
5771:
5031:
This is the type of stochastic convergence that is most similar to
4398:
2993:
2890:{\displaystyle \lim \inf \mathbb {E} f(X_{n})\geq \mathbb {E} f(X)}
186:
102:
of the outcome's distance from a particular value may converge to 0
4267:
Convergence in probability does not imply almost sure convergence.
2576:{\displaystyle \mathbb {P} (X_{n}\leq x)\to \mathbb {P} (X\leq x)}
10523:
gives sufficient conditions for almost sure convergence to imply
3374:{\displaystyle \liminf \mathbb {E} f(X_{n})\geq \mathbb {E} f(X)}
3284:{\displaystyle \limsup \mathbb {E} f(X_{n})\leq \mathbb {E} f(X)}
2425:, the convergence in distribution means that the probability for
11440:
8863:
Convergence in probability implies convergence in distribution:
6959:{\displaystyle {\overset {}{X_{n}\,{\xrightarrow {L^{r}}}\,X.}}}
3175:{\displaystyle \mathbb {P} (X_{n}\in B)\to \mathbb {P} (X\in B)}
1981:{\displaystyle X_{n}\,{\xrightarrow {d}}\,{\mathcal {N}}(0,\,1)}
10959:{\displaystyle \mathbb {E} \rightarrow \mathbb {E} <\infty }
8734:
Convergence in probability implies there exists a sub-sequence
6896:
This type of convergence is often denoted by adding the letter
5759:
Almost sure convergence implies convergence in probability (by
5486:
Almost sure convergence is often denoted by adding the letters
5220:
and the concept of the random variable as a function from Ω to
4407:
4257:
Convergence in probability implies convergence in distribution.
2475:. These random variables converge in distribution to a uniform
2479:(0, 1), whereas their densities do not converge at all.
2328:{\displaystyle \mathbb {E} ^{*}h(X_{n})\to \mathbb {E} \,h(X)}
436:
is a random variable, and all of them are defined on the same
10862:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X}
8356:{\displaystyle Y_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ Y}
8298:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X}
7981:{\displaystyle X_{n}Y_{n}{\xrightarrow {\overset {}{p}}}\ XY}
7699:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ Y}
7641:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{L^{r}}}}\ X}
2346:, that is the expectation of a âsmallest measurable function
11630:
11621:
Creative
Commons Attribution-ShareAlike 3.0 Unported License
8641:
Almost sure convergence implies convergence in probability:
7256:
The converse is not necessarily true, however it is true if
5779:
5770:
The concept of almost sure convergence does not come from a
11423:(2nd ed.). Clarendon Press, Oxford. pp. 271â285.
11320:
Efficient and adaptive estimation for semiparametric models
10541:
3977:
Convergence in probability is denoted by adding the letter
821:
distribution. This convergence is shown in the picture: as
527:
be the fraction of heads after tossing up an unbiased coin
322:
260:{\displaystyle X_{n}={\frac {1}{n}}\sum _{i=1}^{n}Y_{i}\,,}
182:
11317:
11103:
10202:
it is always possible to find a new probability space (Ω,
7298:{\displaystyle {\overset {}{X_{n}\,\xrightarrow {p} \,X}}}
6294:{\displaystyle P(|X_{n}|\geq \varepsilon )={\frac {1}{n}}}
5334:, almost sure convergence can also be defined as follows:
7830:{\displaystyle Y_{n}\ {\xrightarrow {\overset {}{p}}}\ Y}
7779:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ X}
7420:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ Y}
7369:{\displaystyle X_{n}\ {\xrightarrow {\overset {}{p}}}\ X}
5932:. (Note that random variables themselves are functions).
2514:
if and only if any of the following statements are true:
10802:{\displaystyle X_{n}\ \xrightarrow {\overset {}{p}} \ X}
2798:{\displaystyle \mathbb {E} f(X_{n})\to \mathbb {E} f(X)}
2682:{\displaystyle \mathbb {E} f(X_{n})\to \mathbb {E} f(X)}
5011:, ⊠be the daily amounts the charity received from him.
11442:(3rd ed.). HCĂ-tryk, Copenhagen. pp. 18â20.
10544:
8478:
4341:
4300:
1890:
1205:, then this sequence converges in distribution to the
730:
715:
606:
94:
Some less obvious, more theoretical patterns could be
11826:
10974:
10877:
10817:
10766:
10739:{\displaystyle X_{n}{\xrightarrow {\overset {}{P}}}X}
10707:
10539:
10442:
10269:
9923:
9826:
9615:
9579:
9370:
9228:
9078:
8971:
8871:
8775:
8740:
8649:
8476:
8369:
8311:
8253:
8189:
8099:
8048:
7997:
7933:
7843:
7792:
7741:
7712:
7654:
7596:
7567:
7516:
7465:
7433:
7382:
7331:
7262:
7109:
6911:
6879:
6852:
6747:
6707:
6678:
6607:
6587:
6567:
6534:
6470:
6427:
6363:
6327:
6307:
6242:
6202:
6141:
6086:
6053:
5940:
5928:
of a sequence of functions extended to a sequence of
5829:
5628:
5596:
5501:
5340:
5230:
5213:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
5182:
5080:
4947:
4798:
4778:
4751:
4724:
4665:
4638:
4539:
4418:
4280:
4166:
4157:, convergence in probability is defined similarly by
3992:
3770:
3605:
A natural link to convergence in distribution is the
3391:
3323:
3301:
3233:
3211:
3191:
3121:
3099:
3023:
3001:
2925:
2903:
2836:
2814:
2750:
2722:
2702:
2634:
2589:
2522:
2379:
2272:
2172:
2086:
2000:
1930:
1889:
1583:
1555:
1529:
1509:
1467:
1432:
1406:
1373:
1331:
1305:
1285:
1241:
1215:
1168:
1137:
1117:
1090:
1062:
989:
917:
864:
714:
605:
476:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
445:
422:
389:
351:
331:
296:
276:
198:
133:
11527:
11115:
6414:{\displaystyle \sum _{n\geq 1}P(X_{n}=1)\to \infty }
3663:
will converge in probability to the random variable
590:
of the normal distribution. If we shift and rescale
109:
describing the next event grows smaller and smaller.
11509:Grimmett, Geoffrey R.; Stirzaker, David R. (2020).
8963:-th order mean implies convergence in probability:
1131:should be considered is essential. For example, if
1111:The requirement that only the continuity points of
11839:
11347:
11010:
10958:
10861:
10801:
10738:
10672:
10488:
10338:
10076:
9844:
9768:
9597:
9515:
9307:
9171:
9054:
8947:
8851:
8759:
8722:
8626:
8444:
8355:
8297:
8236:
8167:
8085:
8034:
7980:
7911:
7829:
7778:
7724:
7698:
7640:
7579:
7553:
7502:
7445:
7419:
7368:
7297:
7245:
6958:
6885:
6865:
6823:
6715:
6686:
6613:
6593:
6573:
6553:
6520:
6452:
6413:
6346:
6313:
6293:
6228:
6188:
6127:
6072:
6024:
5903:
5743:
5622:, convergence almost surely is defined similarly:
5614:
5552:
5476:
5320:
5212:
5138:
4953:
4931:
4784:
4764:
4737:
4710:
4651:
4615:
4525:
4385:
4326:
4286:
4240:
4109:
3846:
3397:
3373:
3307:
3283:
3217:
3197:
3174:
3105:
3082:
3007:
2984:
2909:
2889:
2820:
2797:
2730:
2708:
2681:
2618:
2575:
2417:
2327:
2193:
2155:
2057:
1980:
1908:
1859:
1561:
1541:
1523:. Thus the convergence of cdfs fails at the point
1515:
1495:
1453:
1418:
1392:
1359:
1317:
1291:
1269:
1227:
1197:
1150:
1123:
1096:
1076:
1046:{\displaystyle \lim _{n\to \infty }F_{n}(x)=F(x),}
1045:
949:
896:
787:be their (normalized) sums. Then according to the
779:
657:
475:
428:
408:
364:
337:
309:
282:
259:
173:
84:An increasing preference towards a certain outcome
11508:
11418:
11252:
5730:
5636:
5447:
5348:
5307:
5238:
5108:
5086:
4443:
3462:Note however that convergence in distribution of
2235:), and we say that a sequence of random elements
12149:
11354:(2nd ed.). John Wiley & Sons. pp.
10167:are almost surely bounded above and below, then
7153:
6749:
6478:
5959:
5849:
5354:
5256:
5093:
4961:. So we do not have convergence in probability.
4571:
4440:
3772:
3324:
3234:
3027:
3024:
2929:
2926:
2840:
2837:
2088:
991:
11456:
1909:{\displaystyle \scriptstyle {\mathcal {L}}_{X}}
6842:-th mean tells us that the expectation of the
2619:{\displaystyle x\mapsto \mathbb {P} (X\leq x)}
1572:Convergence in distribution may be denoted as
486:
54:. The same concepts are known in more general
11646:
11549:
11491:Counterexamples in Probability and Statistics
11488:Romano, Joseph P.; Siegel, Andrew F. (1985).
11465:. Berlin: Springer-Verlag. pp. xii+480.
11396:. New York: Springer Science+Business Media.
11377:. Cambridge, UK: Cambridge University Press.
11264:
11201:
11154:
5921:over which the random variables are defined.
5700:
5659:
5440:
5371:
4518:
4505:
4473:
4446:
4227:
4189:
3833:
3794:
3613:
3566:if and only if the sequence of corresponding
1426:. However, for this limiting random variable
1198:{\displaystyle \left(0,{\frac {1}{n}}\right)}
11611:This article incorporates material from the
11487:
11391:
11276:
11127:
11005:
10975:
7077:â„ 1, implies convergence in probability (by
6548:
6535:
6521:{\displaystyle P(\limsup _{n}\{X_{n}=1\})=1}
6506:
6487:
6447:
6428:
6067:
6054:
25:convergence of sequences of random variables
11590:Stochastic Processes in Engineering Systems
11345:
11336:
10489:{\displaystyle S_{n}=X_{1}+\cdots +X_{n}\,}
6982:The most important cases of convergence in
6189:{\displaystyle P(X_{n}=0)=1-{\frac {1}{n}}}
23:, there exist several different notions of
12070:Vitale's random BrunnâMinkowski inequality
11653:
11639:
10697:A necessary and sufficient condition for
9796:then we wouldn't be able to conclude that
6080:of independent random variables such that
4964:
4386:{\textstyle g(X_{n})\xrightarrow {p} g(X)}
4274:states that for every continuous function
2897:for all nonnegative, continuous functions
2418:{\displaystyle F(a)=\mathbb {P} (X\leq a)}
98:That the series formed by calculating the
11587:
11513:(4th ed.). Oxford University Press.
11035:Proofs of convergence of random variables
10920:
10879:
10608:
10485:
10281:
7287:
7276:
7210:
7169:
6945:
6925:
6765:
6709:
6680:
6128:{\displaystyle P(X_{n}=1)={\frac {1}{n}}}
5780:Sure convergence or pointwise convergence
5724:
5705:
5653:
5630:
5539:
5515:
5342:
5232:
5203:
5082:
4616:{\displaystyle d(X,Y)=\mathbb {E} \left.}
4562:
4467:
4183:
4087:
3788:
3355:
3328:
3265:
3238:
3153:
3123:
3061:
3031:
2963:
2933:
2871:
2844:
2779:
2752:
2724:
2663:
2636:
2597:
2554:
2524:
2396:
2312:
2308:
2275:
2194:{\displaystyle A\subset \mathbb {R} ^{k}}
2181:
2134:
2104:
2045:
1971:
1954:
1941:
1916:is the law (probability distribution) of
1070:
466:
383:Throughout the following, we assume that
253:
11596:
11568:
11554:. New York: Cambridge University Press.
11533:Weak convergence and empirical processes
11437:
11419:Grimmett, G.R.; Stirzaker, D.R. (1992).
10412:. This is a direct implication from the
10379:then it also converges almost surely to
9545:converges in distribution to a constant
9203:converges in distribution to a constant
6229:{\displaystyle 0<\varepsilon <1/2}
3909: > 0 there exists a number
11394:A Modern Approach to Probability Theory
11392:Fristedt, Bert; Gray, Lawrence (1997).
11139:
10507:converges almost surely if and only if
9351:converges in probability to zero, then
492:Examples of convergence in distribution
416:is a sequence of random variables, and
12150:
11597:Zitkovic, Gordan (November 17, 2013).
11372:
11197:
11195:
11193:
11191:
11189:
11187:
11178:
11166:
11133:
6900:over an arrow indicating convergence:
6624:
5490:over an arrow indicating convergence:
5224:, this is equivalent to the statement
3894:is said to converge in probability to
3648:responses. Then (provided there is no
3619:Examples of convergence in probability
3414:states that for a continuous function
11634:
11494:. Great Britain: Chapman & Hall.
10217:= 0, 1, ...} defined on it such that
10195:} which converges in distribution to
5159:, in the sense that events for which
4397:Convergence in probability defines a
3496:imply convergence in distribution of
2338:for all continuous bounded functions
12083:Applications & related
10530:
10375:converges almost completely towards
6902:
6846:-th power of the difference between
6042:
5492:
5332:limit superior of a sequence of sets
4327:{\textstyle X_{n}\xrightarrow {p} X}
3983:
2490:implies convergence in distribution.
1574:
1393:{\displaystyle x\geq {\frac {1}{n}}}
174:{\displaystyle Y_{i},\ i=1,\dots ,n}
12002:Marcinkiewicz interpolation theorem
11350:Convergence of probability measures
11213:
11184:
10432:real independent random variables:
4970:Examples of almost sure convergence
4711:{\displaystyle Y_{n}=(-1)^{n}X_{n}}
3672:Predicting random number generation
950:{\displaystyle F_{1},F_{2},\ldots }
897:{\displaystyle X_{1},X_{2},\ldots }
13:
11928:Symmetric decreasing rearrangement
11832:
11070:Skorokhod's representation theorem
11052:: the question of continuity of a
10953:
10624:
10330:
9358:also converges in distribution to
7317:Provided the probability space is
7163:
6759:
6408:
6016:
5969:
5952:
5859:
5839:
5830:
5718:
5647:
5529:
5523:
5382:
5364:
5266:
5249:
5194:
5186:
5103:
4718:. Notice that the distribution of
4626:
4167:
4081:
3782:
3607:Skorokhod's representation theorem
2098:
1957:
1894:
1836:
1810:
1739:
1689:
1649:
1001:
826:
558:. The subsequent random variables
457:
449:
14:
12174:
10809:, the followings are equivalent
6834:where the operator E denotes the
3711:will converge to the outcomes of
1077:{\displaystyle x\in \mathbb {R} }
910:cumulative distribution functions
11511:Probability and Random Processes
11421:Probability and random processes
11142:Probability: Theory and Examples
11116:van der Vaart & Wellner 1996
10408:also converges almost surely to
7093:-th mean implies convergence in
4745:is equal to the distribution of
3700:converge to the distribution of
1924:is standard normal we can write
975:cumulative distribution function
11619:", which is licensed under the
11282:
11270:
11258:
11246:
11075:The Tweedie convergence theorem
11011:{\displaystyle \{|X_{n}|^{r}\}}
10639:
10635:
10008:
10004:
9700:
9696:
9478:
9474:
9267:
9263:
9124:
9120:
9017:
9013:
8910:
8906:
8812:
8808:
8767:which almost surely converges:
8685:
8681:
7312:
7151:
7147:
5753:
5464:
5458:
5176:). Using the probability space
5042:
4251:
3735:
3594:Convergence in distribution is
3583:to the characteristic function
372:. This result is known as the
120:For example, if the average of
11573:. Cambridge University Press.
11550:van der Vaart, Aad W. (1998).
11216:Probability: A graduate course
11207:
11172:
11160:
11148:
11121:
11109:
11097:
10995:
10979:
10947:
10937:
10928:
10924:
10916:
10913:
10903:
10887:
10883:
10636:
10618:
10612:
10593:
10578:
10312:
10291:
10071:
10059:
10035:
10009:
10005:
9839:
9827:
9763:
9751:
9727:
9701:
9697:
9592:
9580:
9475:
9443:
9415:
9264:
9121:
9014:
8907:
8809:
8754:
8741:
8682:
8603:
8584:
8561:
8507:
7305:(by a more general version of
7237:
7227:
7218:
7214:
7203:
7193:
7177:
7173:
7160:
7148:
6797:
6775:
6756:
6661:) towards the random variable
6509:
6474:
6405:
6402:
6383:
6338:
6275:
6265:
6250:
6246:
6164:
6145:
6109:
6090:
6005:
5999:
5990:
5984:
5966:
5895:
5889:
5880:
5874:
5856:
5715:
5708:
5695:
5689:
5680:
5674:
5609:
5597:
5428:
5424:
5418:
5409:
5403:
5389:
5361:
5302:
5296:
5287:
5281:
5263:
5207:
5183:
5148:This means that the values of
5100:
4926:
4916:
4912:
4903:
4893:
4884:
4880:
4872:
4857:
4853:
4844:
4834:
4806:
4802:
4689:
4679:
4602:
4592:
4578:
4574:
4555:
4543:
4533:or alternately by this metric
4493:
4479:
4434:
4422:
4380:
4374:
4358:
4345:
4232:
4216:
4197:
4078:
3879:is outside the ball of radius
3821:
3800:
3779:
3368:
3362:
3348:
3335:
3278:
3272:
3258:
3245:
3169:
3157:
3149:
3146:
3127:
3077:
3065:
3054:
3035:
2979:
2967:
2956:
2937:
2884:
2878:
2864:
2851:
2792:
2786:
2775:
2772:
2759:
2676:
2670:
2659:
2656:
2643:
2613:
2601:
2593:
2570:
2558:
2550:
2547:
2528:
2412:
2400:
2389:
2383:
2322:
2316:
2304:
2301:
2288:
2150:
2138:
2127:
2108:
2095:
1975:
1962:
1847:
1841:
1831:
1828:
1815:
1793:
1768:
1484:
1478:
1442:
1436:
1348:
1342:
1258:
1252:
1037:
1031:
1022:
1016:
998:
651:
632:
470:
446:
403:
390:
1:
11898:Convergence almost everywhere
11660:
11599:"Lecture 7: Weak Convergence"
11535:. New York: Springer-Verlag.
11375:Real analysis and probability
11346:Billingsley, Patrick (1999).
11337:Billingsley, Patrick (1986).
11322:. New York: Springer-Verlag.
11310:
11253:Grimmett & Stirzaker 2020
11065:Big O in probability notation
11050:Continuous stochastic process
10521:dominated convergence theorem
9785:Note that the condition that
9571:converges in distribution to
9534:converges in distribution to
9333:converges in distribution to
3562:converges in distribution to
3448:converges in distribution to
3429:converges in distribution to
2583:for all continuity points of
2510:converges in distribution to
2488:probability density functions
2449:probability density functions
2367:
853:
181:, all having the same finite
66:
11592:. New York: SpringerâVerlag.
11588:Wong, E.; HĂĄjek, B. (1985).
11571:Probability with Martingales
11463:Probability in Banach spaces
10390:converges in probability to
10224:is equal in distribution to
10156:converges in probability to
10095:converges in probability to
9903:converges in probability to
9877:converges in probability to
9866:converges in probability to
9216:converges in probability to
6716:{\displaystyle \mathbb {E} }
6687:{\displaystyle \mathbb {E} }
5784:To say that the sequence of
3753:towards the random variable
3644:be the average of the first
2731:{\displaystyle \mathbb {E} }
2233:weak convergence of measures
7:
12065:PrĂ©kopaâLeindler inequality
11918:Locally integrable function
11840:{\displaystyle L^{\infty }}
11028:
10686:
10358:converges almost completely
10206:, P) and random variables {
9337:and the difference between
6972:
6636:, we say that the sequence
6462:second Borel Cantelli Lemma
6453:{\displaystyle \{X_{n}=1\}}
5566:
4941:which does not converge to
4123:
3946:(the definition of limit).
2436:is in that range, provided
1873:
487:Convergence in distribution
33:convergence in distribution
10:
12179:
11811:Square-integrable function
11531:; Wellner, Jon A. (1996).
11294:Mathematics Stack Exchange
11085:Continuous mapping theorem
10186:Almost sure representation
6347:{\displaystyle X_{n}\to 0}
5021:However, when we consider
4272:continuous mapping theorem
3870:) be the probability that
3633:be their height, which is
3614:Convergence in probability
3412:continuous mapping theorem
1496:{\displaystyle F_{n}(0)=0}
1360:{\displaystyle F_{n}(x)=1}
1270:{\displaystyle F_{n}(x)=0}
667:converging in distribution
345:, of the random variables
321:(see below) to the common
29:convergence in probability
12163:Convergence (mathematics)
12082:
12060:MinkowskiâSteiner formula
12030:
11992:
11936:
11885:
11819:
11763:
11732:
11668:
11402:10.1007/978-1-4899-2837-5
11218:. Theorem 3.4: Springer.
10514:converges in probability.
10160:and all random variables
6554:{\displaystyle \{X_{n}\}}
6073:{\displaystyle \{X_{n}\}}
5047:To say that the sequence
4993:
4988:
4979:
4974:
3730:weak law of large numbers
3676:
3671:
3628:
3623:
3549:LĂ©vyâs continuity theorem
2067:converges in distribution
682:
677:
519:
514:
501:
496:
374:weak law of large numbers
105:That the variance of the
12043:Isoperimetric inequality
11277:Fristedt & Gray 1997
11238:: CS1 maint: location (
11128:Romano & Siegel 1985
11090:
9881:, then the joint vector
9549:, then the joint vector
5797:) defined over the same
5330:Using the notion of the
5170:have probability 0 (see
3751:converges in probability
3568:characteristic functions
959:converge in distribution
840:probability distribution
575:will all be distributed
12048:BrunnâMinkowski theorem
11339:Probability and Measure
11169:, Chapter 9.2, page 287
11060:Asymptotic distribution
11040:Convergence of measures
8760:{\displaystyle (n_{k})}
4965:Almost sure convergence
1318:{\displaystyle x\leq 0}
409:{\displaystyle (X_{n})}
37:almost sure convergence
11903:Convergence in measure
11841:
11617:Stochastic convergence
11140:Durrett, Rick (2010).
11045:Convergence in measure
11012:
10960:
10863:
10803:
10740:
10674:
10490:
10383:. In other words, if
10340:
10078:
9846:
9770:
9599:
9517:
9309:
9173:
9056:
8949:
8853:
8761:
8724:
8628:
8452:(for any real numbers
8446:
8357:
8299:
8238:
8175:(for any real numbers
8169:
8087:
8036:
7982:
7919:(for any real numbers
7913:
7831:
7780:
7726:
7700:
7642:
7581:
7555:
7504:
7447:
7421:
7370:
7299:
7247:
6960:
6887:
6867:
6825:
6717:
6688:
6615:
6595:
6575:
6555:
6522:
6454:
6415:
6348:
6315:
6295:
6230:
6190:
6129:
6074:
6026:
5924:This is the notion of
5905:
5745:
5616:
5554:
5478:
5322:
5214:
5155:approach the value of
5140:
5035:known from elementary
4955:
4933:
4786:
4766:
4739:
4712:
4659:and a second sequence
4653:
4617:
4527:
4387:
4328:
4288:
4242:
4144:separable metric space
4111:
3848:
3749:} of random variables
3704:, but the outcomes of
3399:
3375:
3309:
3285:
3219:
3199:
3176:
3107:
3084:
3009:
2986:
2911:
2891:
2822:
2799:
2732:
2710:
2683:
2620:
2577:
2482:However, according to
2419:
2342:. Here E* denotes the
2329:
2227:In this case the term
2195:
2157:
2059:
1982:
1910:
1861:
1563:
1543:
1517:
1497:
1455:
1454:{\displaystyle F(0)=1}
1420:
1419:{\displaystyle n>0}
1394:
1361:
1319:
1293:
1271:
1229:
1199:
1152:
1125:
1098:
1078:
1047:
951:
898:
831:
798:approaches the normal
791:, the distribution of
781:
765:
708:random variables. Let
659:
542:Bernoulli distribution
477:
430:
410:
366:
339:
311:
284:
261:
242:
175:
60:stochastic convergence
12017:RieszâFischer theorem
11842:
11801:Polarization identity
11569:Williams, D. (1991).
11552:Asymptotic statistics
11529:van der Vaart, Aad W.
11438:Jacobsen, M. (1992).
11373:Dudley, R.M. (2002).
11013:
10961:
10864:
10804:
10741:
10675:
10491:
10362:almost in probability
10341:
10149:. In other words, if
10079:
9847:
9845:{\displaystyle (X,Y)}
9771:
9600:
9598:{\displaystyle (X,c)}
9518:
9310:
9174:
9057:
8950:
8854:
8762:
8725:
8629:
8447:
8358:
8300:
8239:
8170:
8088:
8037:
7983:
7914:
7832:
7781:
7727:
7701:
7643:
7582:
7556:
7505:
7448:
7422:
7371:
7300:
7248:
6961:
6888:
6868:
6866:{\displaystyle X_{n}}
6826:
6718:
6689:
6616:
6596:
6576:
6561:does not converge to
6556:
6523:
6455:
6416:
6349:
6316:
6296:
6231:
6191:
6130:
6075:
6027:
5926:pointwise convergence
5906:
5746:
5617:
5615:{\displaystyle (S,d)}
5555:
5479:
5323:
5215:
5166:does not converge to
5141:
5033:pointwise convergence
4956:
4934:
4787:
4767:
4765:{\displaystyle X_{n}}
4740:
4738:{\displaystyle Y_{n}}
4713:
4654:
4652:{\displaystyle X_{n}}
4618:
4528:
4388:
4329:
4289:
4243:
4133:For random elements {
4112:
3953:the random variables
3913:(which may depend on
3857:More explicitly, let
3849:
3600:LĂ©vyâProkhorov metric
3400:
3383:lower semi-continuous
3376:
3310:
3293:upper semi-continuous
3286:
3220:
3200:
3177:
3108:
3085:
3010:
2987:
2912:
2892:
2823:
2800:
2733:
2711:
2684:
2621:
2578:
2486:, convergence of the
2420:
2330:
2196:
2158:
2060:
1983:
1911:
1862:
1564:
1544:
1518:
1498:
1456:
1421:
1395:
1362:
1320:
1294:
1272:
1230:
1200:
1153:
1151:{\displaystyle X_{n}}
1126:
1099:
1079:
1048:
969:to a random variable
952:
899:
848:central limit theorem
830:
789:central limit theorem
782:
745:
671:central limit theorem
660:
478:
431:
411:
378:central limit theorem
367:
365:{\displaystyle Y_{i}}
340:
312:
310:{\displaystyle X_{n}}
285:
262:
222:
176:
12158:Stochastic processes
12022:RieszâThorin theorem
11865:Infimum and supremum
11824:
11750:Lebesgue integration
11020:uniformly integrable
10972:
10875:
10815:
10764:
10755:uniformly integrable
10705:
10537:
10440:
10414:BorelâCantelli lemma
10267:
9921:
9824:
9613:
9577:
9368:
9226:
9076:
8969:
8869:
8773:
8738:
8647:
8474:
8367:
8309:
8251:
8187:
8097:
8046:
7995:
7931:
7841:
7790:
7739:
7710:
7652:
7594:
7565:
7514:
7463:
7431:
7380:
7329:
7260:
7107:
7089:â„ 1, convergence in
6909:
6877:
6850:
6745:
6705:
6676:
6629:Given a real number
6605:
6585:
6565:
6532:
6468:
6425:
6361:
6325:
6305:
6240:
6200:
6139:
6084:
6051:
6047:Consider a sequence
5938:
5827:
5765:law of large numbers
5626:
5594:
5499:
5338:
5228:
5180:
5078:
4945:
4796:
4776:
4749:
4722:
4663:
4636:
4537:
4416:
4339:
4298:
4278:
4164:
3990:
3921:) such that for all
3768:
3654:law of large numbers
3389:
3321:
3299:
3231:
3209:
3189:
3119:
3097:
3021:
2999:
2923:
2901:
2834:
2812:
2748:
2720:
2700:
2695:continuous functions
2632:
2587:
2520:
2377:
2270:
2246:converges weakly to
2170:
2084:
1998:
1928:
1887:
1581:
1553:
1527:
1507:
1465:
1430:
1404:
1371:
1329:
1303:
1283:
1239:
1213:
1166:
1135:
1115:
1088:
1060:
987:
915:
862:
712:
603:
599:appropriately, then
544:with expected value
505:uniform distribution
443:
420:
387:
349:
338:{\displaystyle \mu }
329:
294:
274:
196:
131:
52:stochastic processes
11984:Young's convolution
11923:Measurable function
11806:Pythagorean theorem
11796:Parseval's identity
11745:Integrable function
11214:Gut, Allan (2005).
10850:
10791:
10730:
10665:
10568:
10053:
9992:
9949:
9745:
9684:
9641:
9504:
9462:
9396:
9293:
9254:
9157:
9111:
9043:
9004:
8936:
8897:
8841:
8800:
8711:
8672:
8617:
8598:
8579:
8550:
8500:
8421:
8344:
8286:
8222:
8144:
8074:
8023:
7966:
7888:
7818:
7767:
7725:{\displaystyle X=Y}
7687:
7629:
7580:{\displaystyle X=Y}
7542:
7491:
7446:{\displaystyle X=Y}
7408:
7357:
7285:
7138:
7081:). Furthermore, if
7079:Markov's inequality
7069:Convergence in the
6942:
6893:converges to zero.
6625:Convergence in mean
6528:hence the sequence
6301:which converges to
5536:
4971:
4369:
4319:
4052:
4014:
3620:
3581:converges pointwise
3205:of random variable
2807:Lipschitz functions
2231:is preferable (see
2222:empirical processes
1951:
1731:
1693:
1653:
1613:
1542:{\displaystyle x=0}
1228:{\displaystyle X=0}
493:
290:tends to infinity,
12105:Probability theory
12007:Plancherel theorem
11913:Integral transform
11860:Chebyshev distance
11837:
11786:Euclidean distance
11719:Minkowski distance
11623:but not under the
11265:van der Vaart 1998
11202:van der Vaart 1998
11155:van der Vaart 1998
11104:Bickel et al. 1998
11054:stochastic process
11008:
10956:
10859:
10799:
10746:and the sequence (
10736:
10670:
10629:
10486:
10336:
10279:
10074:
9842:
9766:
9595:
9513:
9305:
9169:
9052:
8945:
8849:
8757:
8720:
8624:
8622:
8527:
8442:
8353:
8295:
8234:
8165:
8083:
8032:
7978:
7909:
7827:
7776:
7722:
7696:
7638:
7577:
7551:
7500:
7443:
7417:
7366:
7295:
7243:
7167:
6956:
6883:
6863:
6821:
6763:
6713:
6684:
6611:
6591:
6571:
6551:
6518:
6486:
6450:
6411:
6379:
6344:
6311:
6291:
6226:
6186:
6125:
6070:
6036:probability theory
6022:
5973:
5917:of the underlying
5901:
5863:
5741:
5722:
5612:
5550:
5474:
5368:
5318:
5270:
5210:
5136:
5107:
5064:with probability 1
4969:
4951:
4929:
4782:
4762:
4735:
4708:
4649:
4613:
4523:
4383:
4334:, then also
4324:
4284:
4238:
4107:
4085:
3844:
3786:
3624:Height of a person
3618:
3418:, if the sequence
3395:
3371:
3305:
3281:
3215:
3195:
3172:
3103:
3080:
3005:
2982:
2907:
2887:
2818:
2795:
2728:
2706:
2679:
2616:
2573:
2442:sufficiently large
2415:
2325:
2191:
2153:
2102:
2055:
1978:
1920:. For example, if
1906:
1905:
1857:
1855:
1569:is discontinuous.
1559:
1539:
1513:
1493:
1451:
1416:
1390:
1357:
1315:
1289:
1267:
1225:
1195:
1148:
1121:
1094:
1074:
1043:
1005:
947:
894:
832:
777:
776:
743:
655:
654:
491:
473:
426:
406:
362:
335:
307:
280:
257:
171:
21:probability theory
12145:
12144:
12078:
12077:
11893:Almost everywhere
11678: &
11580:978-0-521-40605-5
11561:978-0-521-49603-2
11542:978-0-387-94640-5
11520:978-0-198-84760-1
11501:978-0-412-98901-8
11472:978-3-540-52013-9
11459:Talagrand, Michel
11449:978-87-91180-71-2
11430:978-0-19-853665-9
11411:978-1-4899-2837-5
11384:978-0-521-80972-6
11365:978-0-471-19745-4
11329:978-0-387-98473-5
11225:978-0-387-22833-4
11080:Slutsky's theorem
10855:
10851:
10849:
10848:
10830:
10795:
10792:
10790:
10789:
10779:
10731:
10729:
10728:
10694:
10693:
10666:
10569:
10567:
10566:
10565:
10350:then we say that
10270:
10058:
10054:
10052:
10051:
10040:
10003:
9997:
9993:
9991:
9990:
9979:
9966:
9963:
9954:
9950:
9948:
9947:
9936:
9750:
9746:
9744:
9743:
9732:
9695:
9689:
9685:
9683:
9682:
9671:
9658:
9655:
9646:
9642:
9640:
9639:
9628:
9509:
9505:
9503:
9502:
9491:
9473:
9467:
9463:
9461:
9460:
9449:
9413:
9410:
9401:
9397:
9395:
9394:
9383:
9298:
9294:
9292:
9291:
9280:
9259:
9255:
9253:
9252:
9241:
9162:
9158:
9156:
9155:
9137:
9116:
9112:
9110:
9109:
9091:
9048:
9044:
9042:
9041:
9030:
9009:
9005:
9003:
9002:
8984:
8941:
8937:
8935:
8934:
8923:
8902:
8898:
8896:
8895:
8884:
8845:
8842:
8840:
8832:
8804:
8801:
8799:
8798:
8788:
8716:
8712:
8710:
8709:
8698:
8677:
8673:
8671:
8662:
8618:
8599:
8580:
8578:
8551:
8549:
8548:
8506:
8501:
8499:
8498:
8426:
8422:
8420:
8419:
8401:
8349:
8345:
8343:
8342:
8324:
8291:
8287:
8285:
8284:
8266:
8227:
8223:
8221:
8220:
8219:
8149:
8145:
8143:
8142:
8141:
8131:
8079:
8075:
8073:
8072:
8071:
8061:
8028:
8024:
8022:
8021:
8020:
8010:
7971:
7967:
7965:
7964:
7893:
7889:
7887:
7886:
7875:
7823:
7819:
7817:
7816:
7805:
7772:
7768:
7766:
7765:
7754:
7692:
7688:
7686:
7685:
7667:
7634:
7630:
7628:
7627:
7609:
7547:
7543:
7541:
7540:
7539:
7529:
7496:
7492:
7490:
7489:
7488:
7478:
7413:
7409:
7407:
7406:
7395:
7362:
7358:
7356:
7355:
7344:
7293:
7292:
7286:
7152:
7145:
7144:
7139:
7059:in quadratic mean
7046:= 2, we say that
7009:= 1, we say that
6980:
6979:
6954:
6953:
6943:
6886:{\displaystyle X}
6838:. Convergence in
6748:
6614:{\displaystyle 1}
6594:{\displaystyle 0}
6574:{\displaystyle 0}
6477:
6460:are independent,
6364:
6314:{\displaystyle 0}
6289:
6184:
6123:
5958:
5919:probability space
5848:
5847:
5799:probability space
5707:
5574:
5573:
5548:
5547:
5537:
5462:
5353:
5255:
5092:
5060:almost everywhere
5029:
5028:
5023:any finite number
4954:{\displaystyle 0}
4785:{\displaystyle n}
4465:
4370:
4320:
4287:{\displaystyle g}
4131:
4130:
4070:
4068:
4065:
4056:
4053:
4043:
4030:
4027:
4018:
4015:
4005:
3942:) <
3883:centered at
3771:
3719:
3718:
3398:{\displaystyle f}
3308:{\displaystyle f}
3218:{\displaystyle X}
3198:{\displaystyle B}
3106:{\displaystyle F}
3008:{\displaystyle G}
2910:{\displaystyle f}
2821:{\displaystyle f}
2805:for all bounded,
2709:{\displaystyle f}
2497:portmanteau lemma
2484:ScheffĂ©âs theorem
2344:outer expectation
2087:
1952:
1881:
1880:
1807:
1804:
1782:
1779:
1735:
1732:
1722:
1709:
1706:
1697:
1694:
1682:
1669:
1666:
1657:
1654:
1642:
1629:
1626:
1617:
1614:
1604:
1562:{\displaystyle F}
1516:{\displaystyle n}
1388:
1292:{\displaystyle n}
1188:
1124:{\displaystyle F}
1097:{\displaystyle F}
1056:for every number
990:
836:
835:
741:
740:
630:
626:
438:probability space
429:{\displaystyle X}
283:{\displaystyle n}
220:
149:
127:random variables
12170:
12095:Fourier analysis
12053:Milman's reverse
12036:
12034:Lebesgue measure
12028:
12027:
12012:RiemannâLebesgue
11855:Bounded function
11846:
11844:
11843:
11838:
11836:
11835:
11755:Taxicab geometry
11710:Measurable space
11655:
11648:
11641:
11632:
11631:
11605:
11603:
11593:
11584:
11565:
11546:
11524:
11505:
11484:
11457:Ledoux, Michel;
11453:
11434:
11415:
11388:
11369:
11353:
11342:
11333:
11304:
11303:
11301:
11300:
11286:
11280:
11274:
11268:
11262:
11256:
11250:
11244:
11243:
11237:
11229:
11211:
11205:
11199:
11182:
11176:
11170:
11164:
11158:
11152:
11146:
11145:
11137:
11131:
11125:
11119:
11113:
11107:
11101:
11017:
11015:
11014:
11009:
11004:
11003:
10998:
10992:
10991:
10982:
10965:
10963:
10962:
10957:
10946:
10945:
10940:
10931:
10923:
10912:
10911:
10906:
10900:
10899:
10890:
10882:
10868:
10866:
10865:
10860:
10853:
10852:
10847:
10846:
10837:
10832:
10828:
10827:
10826:
10808:
10806:
10805:
10800:
10793:
10785:
10780:
10777:
10776:
10775:
10745:
10743:
10742:
10737:
10732:
10724:
10719:
10717:
10716:
10688:
10679:
10677:
10676:
10671:
10664:
10663:
10650:
10649:
10648:
10634:
10630:
10611:
10596:
10591:
10590:
10581:
10563:
10562:
10557:
10556:
10555:
10531:
10513:
10506:
10495:
10493:
10492:
10487:
10484:
10483:
10465:
10464:
10452:
10451:
10427:
10407:
10400:
10389:
10374:
10356:
10345:
10343:
10342:
10337:
10326:
10322:
10315:
10304:
10303:
10294:
10284:
10278:
10237:
10230:
10173:
10166:
10155:
10148:
10133:
10118:
10112:
10094:
10083:
10081:
10080:
10075:
10056:
10055:
10047:
10042:
10038:
10034:
10033:
10021:
10020:
10001:
9995:
9994:
9986:
9981:
9977:
9976:
9975:
9964:
9961:
9952:
9951:
9943:
9938:
9934:
9933:
9932:
9914:
9902:
9853:
9851:
9849:
9848:
9843:
9817:
9791:
9782:
9775:
9773:
9772:
9767:
9748:
9747:
9739:
9734:
9730:
9726:
9725:
9713:
9712:
9693:
9687:
9686:
9678:
9673:
9669:
9668:
9667:
9656:
9653:
9644:
9643:
9635:
9630:
9626:
9625:
9624:
9606:
9604:
9602:
9601:
9596:
9570:
9533:
9522:
9520:
9519:
9514:
9507:
9506:
9498:
9493:
9489:
9488:
9487:
9471:
9465:
9464:
9456:
9451:
9447:
9446:
9441:
9440:
9428:
9427:
9418:
9411:
9408:
9399:
9398:
9390:
9385:
9381:
9380:
9379:
9332:
9321:
9314:
9312:
9311:
9306:
9296:
9295:
9287:
9282:
9278:
9277:
9276:
9257:
9256:
9248:
9243:
9239:
9238:
9237:
9189:
9178:
9176:
9175:
9170:
9160:
9159:
9154:
9153:
9144:
9139:
9135:
9134:
9133:
9114:
9113:
9108:
9107:
9098:
9093:
9089:
9088:
9087:
9061:
9059:
9058:
9053:
9046:
9045:
9037:
9032:
9028:
9027:
9026:
9007:
9006:
9001:
9000:
8991:
8986:
8982:
8981:
8980:
8954:
8952:
8951:
8946:
8939:
8938:
8930:
8925:
8921:
8920:
8919:
8900:
8899:
8891:
8886:
8882:
8881:
8880:
8858:
8856:
8855:
8850:
8843:
8838:
8833:
8830:
8829:
8828:
8827:
8826:
8802:
8794:
8789:
8786:
8785:
8784:
8766:
8764:
8763:
8758:
8753:
8752:
8729:
8727:
8726:
8721:
8714:
8713:
8705:
8700:
8696:
8695:
8694:
8675:
8674:
8669:
8664:
8660:
8659:
8658:
8633:
8631:
8630:
8625:
8623:
8619:
8609:
8600:
8590:
8581:
8576:
8571:
8566:
8565:
8559:
8558:
8555:
8554:
8552:
8547:
8546:
8537:
8532:
8528:
8526:
8502:
8497:
8496:
8487:
8482:
8459:
8455:
8451:
8449:
8448:
8443:
8424:
8423:
8418:
8417:
8408:
8403:
8399:
8398:
8397:
8382:
8381:
8362:
8360:
8359:
8354:
8347:
8346:
8341:
8340:
8331:
8326:
8322:
8321:
8320:
8304:
8302:
8301:
8296:
8289:
8288:
8283:
8282:
8273:
8268:
8264:
8263:
8262:
8243:
8241:
8240:
8235:
8225:
8224:
8217:
8216:
8211:
8209:
8208:
8199:
8198:
8182:
8178:
8174:
8172:
8171:
8166:
8147:
8146:
8139:
8138:
8133:
8129:
8128:
8127:
8112:
8111:
8092:
8090:
8089:
8084:
8077:
8076:
8069:
8068:
8063:
8059:
8058:
8057:
8041:
8039:
8038:
8033:
8026:
8025:
8018:
8017:
8012:
8008:
8007:
8006:
7987:
7985:
7984:
7979:
7969:
7968:
7960:
7955:
7953:
7952:
7943:
7942:
7926:
7922:
7918:
7916:
7915:
7910:
7891:
7890:
7882:
7877:
7873:
7872:
7871:
7856:
7855:
7836:
7834:
7833:
7828:
7821:
7820:
7812:
7807:
7803:
7802:
7801:
7785:
7783:
7782:
7777:
7770:
7769:
7761:
7756:
7752:
7751:
7750:
7731:
7729:
7728:
7723:
7705:
7703:
7702:
7697:
7690:
7689:
7684:
7683:
7674:
7669:
7665:
7664:
7663:
7647:
7645:
7644:
7639:
7632:
7631:
7626:
7625:
7616:
7611:
7607:
7606:
7605:
7586:
7584:
7583:
7578:
7560:
7558:
7557:
7552:
7545:
7544:
7537:
7536:
7531:
7527:
7526:
7525:
7509:
7507:
7506:
7501:
7494:
7493:
7486:
7485:
7480:
7476:
7475:
7474:
7452:
7450:
7449:
7444:
7426:
7424:
7423:
7418:
7411:
7410:
7402:
7397:
7393:
7392:
7391:
7375:
7373:
7372:
7367:
7360:
7359:
7351:
7346:
7342:
7341:
7340:
7304:
7302:
7301:
7296:
7294:
7291:
7277:
7275:
7274:
7264:
7252:
7250:
7249:
7244:
7236:
7235:
7230:
7221:
7213:
7202:
7201:
7196:
7190:
7189:
7180:
7172:
7166:
7146:
7143:
7137:
7136:
7123:
7122:
7121:
7111:
7052:
7033:
7015:
6996:
6974:
6965:
6963:
6962:
6957:
6955:
6952:
6944:
6941:
6940:
6927:
6924:
6923:
6913:
6903:
6892:
6890:
6889:
6884:
6872:
6870:
6869:
6864:
6862:
6861:
6845:
6841:
6830:
6828:
6827:
6822:
6811:
6807:
6806:
6805:
6800:
6788:
6787:
6778:
6768:
6762:
6733:
6722:
6720:
6719:
6714:
6712:
6693:
6691:
6690:
6685:
6683:
6671:absolute moments
6668:
6642:
6635:
6620:
6618:
6617:
6612:
6601:has probability
6600:
6598:
6597:
6592:
6580:
6578:
6577:
6572:
6560:
6558:
6557:
6552:
6547:
6546:
6527:
6525:
6524:
6519:
6499:
6498:
6485:
6459:
6457:
6456:
6451:
6440:
6439:
6420:
6418:
6417:
6412:
6395:
6394:
6378:
6354:in probability.
6353:
6351:
6350:
6345:
6337:
6336:
6320:
6318:
6317:
6312:
6300:
6298:
6297:
6292:
6290:
6282:
6268:
6263:
6262:
6253:
6235:
6233:
6232:
6227:
6222:
6195:
6193:
6192:
6187:
6185:
6177:
6157:
6156:
6134:
6132:
6131:
6126:
6124:
6116:
6102:
6101:
6079:
6077:
6076:
6071:
6066:
6065:
6031:
6029:
6028:
6023:
6012:
6008:
5983:
5982:
5972:
5930:random variables
5910:
5908:
5907:
5902:
5873:
5872:
5862:
5845:
5786:random variables
5750:
5748:
5747:
5742:
5734:
5733:
5723:
5721:
5704:
5703:
5673:
5672:
5663:
5662:
5640:
5639:
5633:
5621:
5619:
5618:
5613:
5568:
5559:
5557:
5556:
5551:
5549:
5546:
5538:
5535:
5517:
5514:
5513:
5503:
5493:
5483:
5481:
5480:
5475:
5463:
5460:
5451:
5450:
5444:
5443:
5431:
5402:
5401:
5392:
5375:
5374:
5367:
5352:
5351:
5345:
5327:
5325:
5324:
5319:
5311:
5310:
5280:
5279:
5269:
5242:
5241:
5235:
5219:
5217:
5216:
5211:
5206:
5198:
5197:
5165:
5154:
5145:
5143:
5142:
5137:
5129:
5125:
5118:
5117:
5106:
5085:
5053:
4972:
4968:
4960:
4958:
4957:
4952:
4938:
4936:
4935:
4930:
4919:
4911:
4910:
4883:
4875:
4870:
4869:
4860:
4837:
4832:
4831:
4819:
4818:
4809:
4791:
4789:
4788:
4783:
4771:
4769:
4768:
4763:
4761:
4760:
4744:
4742:
4741:
4736:
4734:
4733:
4717:
4715:
4714:
4709:
4707:
4706:
4697:
4696:
4675:
4674:
4658:
4656:
4655:
4650:
4648:
4647:
4622:
4620:
4619:
4614:
4609:
4605:
4595:
4581:
4565:
4532:
4530:
4529:
4524:
4522:
4521:
4509:
4508:
4496:
4482:
4477:
4476:
4470:
4463:
4450:
4449:
4394:
4392:
4390:
4389:
4384:
4361:
4357:
4356:
4333:
4331:
4330:
4325:
4311:
4310:
4309:
4293:
4291:
4290:
4285:
4247:
4245:
4244:
4239:
4231:
4230:
4209:
4208:
4193:
4192:
4186:
4156:
4125:
4116:
4114:
4113:
4108:
4097:
4096:
4086:
4084:
4066:
4063:
4054:
4044:
4041:
4040:
4039:
4028:
4025:
4016:
4006:
4003:
4002:
4001:
3984:
3904:
3893:
3853:
3851:
3850:
3845:
3837:
3836:
3824:
3813:
3812:
3803:
3798:
3797:
3791:
3785:
3714:
3710:
3703:
3699:
3692:
3688:
3680:
3666:
3662:
3650:systematic error
3647:
3643:
3632:
3621:
3617:
3590:
3586:
3579:
3565:
3561:
3542:
3538:
3523:
3513:
3492:does in general
3491:
3487:
3476:
3472:
3458:
3447:
3432:
3428:
3417:
3404:
3402:
3401:
3396:
3380:
3378:
3377:
3372:
3358:
3347:
3346:
3331:
3314:
3312:
3311:
3306:
3290:
3288:
3287:
3282:
3268:
3257:
3256:
3241:
3224:
3222:
3221:
3216:
3204:
3202:
3201:
3196:
3181:
3179:
3178:
3173:
3156:
3139:
3138:
3126:
3112:
3110:
3109:
3104:
3089:
3087:
3086:
3081:
3064:
3047:
3046:
3034:
3014:
3012:
3011:
3006:
2991:
2989:
2988:
2983:
2966:
2949:
2948:
2936:
2916:
2914:
2913:
2908:
2896:
2894:
2893:
2888:
2874:
2863:
2862:
2847:
2827:
2825:
2824:
2819:
2804:
2802:
2801:
2796:
2782:
2771:
2770:
2755:
2737:
2735:
2734:
2729:
2727:
2715:
2713:
2712:
2707:
2688:
2686:
2685:
2680:
2666:
2655:
2654:
2639:
2625:
2623:
2622:
2617:
2600:
2582:
2580:
2579:
2574:
2557:
2540:
2539:
2527:
2513:
2509:
2474:
2439:
2435:
2431:
2424:
2422:
2421:
2416:
2399:
2363:
2349:
2341:
2334:
2332:
2331:
2326:
2311:
2300:
2299:
2284:
2283:
2278:
2262:
2249:
2245:
2229:weak convergence
2208:
2200:
2198:
2197:
2192:
2190:
2189:
2184:
2162:
2160:
2159:
2154:
2137:
2120:
2119:
2107:
2101:
2076:
2072:
2064:
2062:
2061:
2056:
2054:
2053:
2048:
2039:
2035:
2028:
2027:
2015:
2014:
1987:
1985:
1984:
1979:
1961:
1960:
1953:
1943:
1940:
1939:
1923:
1919:
1915:
1913:
1912:
1907:
1904:
1903:
1898:
1897:
1875:
1866:
1864:
1863:
1858:
1856:
1840:
1839:
1827:
1826:
1814:
1813:
1805:
1802:
1792:
1791:
1780:
1777:
1767:
1766:
1756:
1749:
1748:
1743:
1742:
1733:
1723:
1720:
1719:
1718:
1707:
1704:
1695:
1692:
1683:
1680:
1679:
1678:
1667:
1664:
1655:
1652:
1643:
1640:
1639:
1638:
1627:
1624:
1615:
1605:
1602:
1601:
1600:
1589:
1575:
1568:
1566:
1565:
1560:
1548:
1546:
1545:
1540:
1522:
1520:
1519:
1514:
1502:
1500:
1499:
1494:
1477:
1476:
1460:
1458:
1457:
1452:
1425:
1423:
1422:
1417:
1399:
1397:
1396:
1391:
1389:
1381:
1366:
1364:
1363:
1358:
1341:
1340:
1324:
1322:
1321:
1316:
1298:
1296:
1295:
1290:
1276:
1274:
1273:
1268:
1251:
1250:
1234:
1232:
1231:
1226:
1209:random variable
1204:
1202:
1201:
1196:
1194:
1190:
1189:
1181:
1158:are distributed
1157:
1155:
1154:
1149:
1147:
1146:
1130:
1128:
1127:
1122:
1103:
1101:
1100:
1095:
1083:
1081:
1080:
1075:
1073:
1052:
1050:
1049:
1044:
1015:
1014:
1004:
979:
972:
956:
954:
953:
948:
940:
939:
927:
926:
906:random variables
903:
901:
900:
895:
887:
886:
874:
873:
824:
820:
818:
816:
815:
812:
809:
797:
786:
784:
783:
778:
775:
774:
764:
759:
744:
742:
736:
732:
725:
724:
707:
693:
664:
662:
661:
656:
644:
643:
631:
622:
621:
616:
615:
598:
585:
574:
557:
550:
539:
530:
526:
494:
490:
482:
480:
479:
474:
469:
461:
460:
435:
433:
432:
427:
415:
413:
412:
407:
402:
401:
371:
369:
368:
363:
361:
360:
344:
342:
341:
336:
316:
314:
313:
308:
306:
305:
289:
287:
286:
281:
266:
264:
263:
258:
252:
251:
241:
236:
221:
213:
208:
207:
180:
178:
177:
172:
147:
143:
142:
12178:
12177:
12173:
12172:
12171:
12169:
12168:
12167:
12148:
12147:
12146:
12141:
12074:
12031:
12026:
11988:
11964:HausdorffâYoung
11944:BabenkoâBeckner
11932:
11881:
11831:
11827:
11825:
11822:
11821:
11815:
11759:
11728:
11724:Sequence spaces
11664:
11659:
11608:
11601:
11581:
11562:
11543:
11521:
11502:
11473:
11450:
11431:
11412:
11385:
11366:
11330:
11313:
11308:
11307:
11298:
11296:
11288:
11287:
11283:
11275:
11271:
11263:
11259:
11251:
11247:
11231:
11230:
11226:
11212:
11208:
11200:
11185:
11177:
11173:
11165:
11161:
11153:
11149:
11138:
11134:
11126:
11122:
11114:
11110:
11106:, A.8, page 475
11102:
11098:
11093:
11031:
10999:
10994:
10993:
10987:
10983:
10978:
10973:
10970:
10969:
10941:
10936:
10935:
10927:
10919:
10907:
10902:
10901:
10895:
10891:
10886:
10878:
10876:
10873:
10872:
10842:
10838:
10831:
10822:
10818:
10816:
10813:
10812:
10771:
10767:
10765:
10762:
10761:
10751:
10718:
10712:
10708:
10706:
10703:
10702:
10701:convergence is
10659:
10655:
10644:
10640:
10628:
10627:
10607:
10604:
10603:
10592:
10586:
10582:
10577:
10574:
10573:
10551:
10547:
10543:
10540:
10538:
10535:
10534:
10512:
10508:
10505:
10501:
10479:
10475:
10460:
10456:
10447:
10443:
10441:
10438:
10437:
10426:
10422:
10406:
10402:
10395:
10388:
10384:
10373:
10369:
10355:
10351:
10311:
10299:
10295:
10290:
10289:
10285:
10280:
10274:
10268:
10265:
10264:
10251:
10243:
10232:
10229:
10225:
10222:
10211:
10201:
10193:
10172:
10168:
10165:
10161:
10154:
10150:
10143:
10132:
10128:
10110:
10105:
10100:
10093:
10089:
10041:
10029:
10025:
10016:
10012:
9980:
9971:
9967:
9937:
9928:
9924:
9922:
9919:
9918:
9904:
9900:
9891:
9882:
9875:
9864:
9825:
9822:
9821:
9819:
9815:
9806:
9797:
9790:
9786:
9776:
9733:
9721:
9717:
9708:
9704:
9672:
9663:
9659:
9629:
9620:
9616:
9614:
9611:
9610:
9578:
9575:
9574:
9572:
9568:
9559:
9550:
9543:
9532:
9528:
9492:
9483:
9479:
9450:
9442:
9436:
9432:
9423:
9419:
9414:
9384:
9375:
9371:
9369:
9366:
9365:
9356:
9349:
9342:
9331:
9327:
9315:
9281:
9272:
9268:
9242:
9233:
9229:
9227:
9224:
9223:
9215:
9202:
9179:
9149:
9145:
9138:
9129:
9125:
9103:
9099:
9092:
9083:
9079:
9077:
9074:
9073:
9066:Convergence in
9031:
9022:
9018:
8996:
8992:
8985:
8976:
8972:
8970:
8967:
8966:
8959:Convergence in
8924:
8915:
8911:
8885:
8876:
8872:
8870:
8867:
8866:
8822:
8818:
8817:
8813:
8780:
8776:
8774:
8771:
8770:
8748:
8744:
8739:
8736:
8735:
8699:
8690:
8686:
8663:
8654:
8650:
8648:
8645:
8644:
8621:
8620:
8608:
8606:
8601:
8589:
8587:
8582:
8570:
8567:
8564:
8556:
8553:
8542:
8538:
8531:
8529:
8510:
8505:
8503:
8492:
8488:
8481:
8477:
8475:
8472:
8471:
8457:
8453:
8413:
8409:
8402:
8393:
8389:
8377:
8373:
8368:
8365:
8364:
8336:
8332:
8325:
8316:
8312:
8310:
8307:
8306:
8278:
8274:
8267:
8258:
8254:
8252:
8249:
8248:
8210:
8204:
8200:
8194:
8190:
8188:
8185:
8184:
8180:
8176:
8132:
8123:
8119:
8107:
8103:
8098:
8095:
8094:
8062:
8053:
8049:
8047:
8044:
8043:
8011:
8002:
7998:
7996:
7993:
7992:
7954:
7948:
7944:
7938:
7934:
7932:
7929:
7928:
7924:
7920:
7876:
7867:
7863:
7851:
7847:
7842:
7839:
7838:
7806:
7797:
7793:
7791:
7788:
7787:
7755:
7746:
7742:
7740:
7737:
7736:
7711:
7708:
7707:
7679:
7675:
7668:
7659:
7655:
7653:
7650:
7649:
7621:
7617:
7610:
7601:
7597:
7595:
7592:
7591:
7566:
7563:
7562:
7530:
7521:
7517:
7515:
7512:
7511:
7479:
7470:
7466:
7464:
7461:
7460:
7432:
7429:
7428:
7396:
7387:
7383:
7381:
7378:
7377:
7345:
7336:
7332:
7330:
7327:
7326:
7315:
7307:Scheffé's lemma
7270:
7266:
7265:
7263:
7261:
7258:
7257:
7231:
7226:
7225:
7217:
7209:
7197:
7192:
7191:
7185:
7181:
7176:
7168:
7156:
7132:
7128:
7117:
7113:
7112:
7110:
7108:
7105:
7104:
7100:Additionally,
7055:in mean square
7051:
7047:
7032:
7028:
7014:
7010:
6995:
6991:
6936:
6932:
6926:
6919:
6915:
6914:
6912:
6910:
6907:
6906:
6878:
6875:
6874:
6857:
6853:
6851:
6848:
6847:
6843:
6839:
6801:
6796:
6795:
6783:
6779:
6774:
6773:
6769:
6764:
6752:
6746:
6743:
6742:
6732:
6728:
6708:
6706:
6703:
6702:
6699:
6679:
6677:
6674:
6673:
6666:
6641:
6637:
6630:
6627:
6606:
6603:
6602:
6586:
6583:
6582:
6566:
6563:
6562:
6542:
6538:
6533:
6530:
6529:
6494:
6490:
6481:
6469:
6466:
6465:
6435:
6431:
6426:
6423:
6422:
6421:and the events
6390:
6386:
6368:
6362:
6359:
6358:
6332:
6328:
6326:
6323:
6322:
6306:
6303:
6302:
6281:
6264:
6258:
6254:
6249:
6241:
6238:
6237:
6218:
6201:
6198:
6197:
6176:
6152:
6148:
6140:
6137:
6136:
6115:
6097:
6093:
6085:
6082:
6081:
6061:
6057:
6052:
6049:
6048:
6045:
6043:Counterexamples
5978:
5974:
5962:
5945:
5941:
5939:
5936:
5935:
5913:where Ω is the
5868:
5864:
5852:
5828:
5825:
5824:
5796:
5782:
5756:
5729:
5728:
5711:
5706:
5699:
5698:
5668:
5664:
5658:
5657:
5635:
5634:
5629:
5627:
5624:
5623:
5595:
5592:
5591:
5585:
5578:random elements
5522:
5516:
5509:
5505:
5504:
5502:
5500:
5497:
5496:
5459:
5446:
5445:
5439:
5438:
5427:
5397:
5393:
5388:
5370:
5369:
5357:
5347:
5346:
5341:
5339:
5336:
5335:
5306:
5305:
5275:
5271:
5259:
5237:
5236:
5231:
5229:
5226:
5225:
5202:
5193:
5192:
5181:
5178:
5177:
5164:
5160:
5153:
5149:
5113:
5109:
5096:
5091:
5087:
5081:
5079:
5076:
5075:
5052:
5048:
5045:
5020:
5019:
5013:
5012:
5010:
5003:
4996:
4995:
4967:
4946:
4943:
4942:
4915:
4906:
4902:
4879:
4871:
4865:
4861:
4856:
4833:
4827:
4823:
4814:
4810:
4805:
4797:
4794:
4793:
4777:
4774:
4773:
4756:
4752:
4750:
4747:
4746:
4729:
4725:
4723:
4720:
4719:
4702:
4698:
4692:
4688:
4670:
4666:
4664:
4661:
4660:
4643:
4639:
4637:
4634:
4633:
4629:
4627:Counterexamples
4591:
4577:
4570:
4566:
4561:
4538:
4535:
4534:
4517:
4516:
4504:
4503:
4492:
4478:
4472:
4471:
4466:
4445:
4444:
4417:
4414:
4413:
4352:
4348:
4340:
4337:
4336:
4335:
4305:
4301:
4299:
4296:
4295:
4279:
4276:
4275:
4254:
4226:
4225:
4204:
4200:
4188:
4187:
4182:
4165:
4162:
4161:
4146:
4141:
4092:
4088:
4074:
4069:
4035:
4031:
3997:
3993:
3991:
3988:
3987:
3965:
3937:
3899:
3892:
3888:
3878:
3865:
3832:
3831:
3820:
3808:
3804:
3799:
3793:
3792:
3787:
3775:
3769:
3766:
3765:
3748:
3738:
3712:
3709:
3705:
3701:
3698:
3694:
3690:
3687:
3683:
3678:
3664:
3661:
3657:
3656:, the sequence
3645:
3642:
3638:
3630:
3616:
3588:
3584:
3576:
3570:
3563:
3558:
3552:
3551:: The sequence
3540:
3535:
3531:
3525:
3515:
3510:
3503:
3497:
3489:
3484:
3478:
3474:
3469:
3463:
3449:
3444:
3434:
3430:
3425:
3419:
3415:
3390:
3387:
3386:
3354:
3342:
3338:
3327:
3322:
3319:
3318:
3300:
3297:
3296:
3264:
3252:
3248:
3237:
3232:
3229:
3228:
3210:
3207:
3206:
3190:
3187:
3186:
3184:continuity sets
3152:
3134:
3130:
3122:
3120:
3117:
3116:
3098:
3095:
3094:
3060:
3042:
3038:
3030:
3022:
3019:
3018:
3000:
2997:
2996:
2962:
2944:
2940:
2932:
2924:
2921:
2920:
2902:
2899:
2898:
2870:
2858:
2854:
2843:
2835:
2832:
2831:
2813:
2810:
2809:
2778:
2766:
2762:
2751:
2749:
2746:
2745:
2723:
2721:
2718:
2717:
2701:
2698:
2697:
2662:
2650:
2646:
2635:
2633:
2630:
2629:
2596:
2588:
2585:
2584:
2553:
2535:
2531:
2523:
2521:
2518:
2517:
2511:
2506:
2500:
2473:
2457:
2452:
2437:
2433:
2430:
2426:
2395:
2378:
2375:
2374:
2370:
2360:
2351:
2350:that dominates
2347:
2339:
2307:
2295:
2291:
2279:
2274:
2273:
2271:
2268:
2267:
2256:
2251:
2247:
2242:
2236:
2214:random elements
2206:
2185:
2180:
2179:
2171:
2168:
2167:
2133:
2115:
2111:
2103:
2091:
2085:
2082:
2081:
2074:
2070:
2049:
2044:
2043:
2023:
2019:
2010:
2006:
2005:
2001:
1999:
1996:
1995:
1956:
1955:
1942:
1935:
1931:
1929:
1926:
1925:
1921:
1917:
1899:
1893:
1892:
1891:
1888:
1885:
1884:
1854:
1853:
1835:
1834:
1822:
1818:
1809:
1808:
1787:
1783:
1762:
1758:
1754:
1753:
1744:
1738:
1737:
1736:
1714:
1710:
1688:
1674:
1670:
1648:
1634:
1630:
1596:
1592:
1590:
1588:
1584:
1582:
1579:
1578:
1554:
1551:
1550:
1528:
1525:
1524:
1508:
1505:
1504:
1472:
1468:
1466:
1463:
1462:
1431:
1428:
1427:
1405:
1402:
1401:
1380:
1372:
1369:
1368:
1336:
1332:
1330:
1327:
1326:
1304:
1301:
1300:
1284:
1281:
1280:
1246:
1242:
1240:
1237:
1236:
1214:
1211:
1210:
1180:
1173:
1169:
1167:
1164:
1163:
1142:
1138:
1136:
1133:
1132:
1116:
1113:
1112:
1089:
1086:
1085:
1069:
1061:
1058:
1057:
1010:
1006:
994:
988:
985:
984:
977:
970:
967:converge in law
963:converge weakly
935:
931:
922:
918:
916:
913:
912:
904:of real-valued
882:
878:
869:
865:
863:
860:
859:
856:
822:
813:
810:
807:
806:
804:
799:
796:
792:
770:
766:
760:
749:
731:
729:
720:
716:
713:
710:
709:
702:
690:
684:
678:Graphic example
639:
635:
620:
611:
607:
604:
601:
600:
596:
591:
583:
581:
580:
572:
565:
559:
552:
545:
538:
532:
528:
525:
521:
509:
508:
489:
465:
456:
455:
444:
441:
440:
421:
418:
417:
397:
393:
388:
385:
384:
356:
352:
350:
347:
346:
330:
327:
326:
301:
297:
295:
292:
291:
275:
272:
271:
247:
243:
237:
226:
212:
203:
199:
197:
194:
193:
138:
134:
132:
129:
128:
107:random variable
69:
17:
12:
11:
5:
12176:
12166:
12165:
12160:
12143:
12142:
12140:
12139:
12138:
12137:
12132:
12122:
12117:
12112:
12107:
12102:
12097:
12092:
12086:
12084:
12080:
12079:
12076:
12075:
12073:
12072:
12067:
12062:
12057:
12056:
12055:
12045:
12039:
12037:
12025:
12024:
12019:
12014:
12009:
12004:
11998:
11996:
11990:
11989:
11987:
11986:
11981:
11976:
11971:
11966:
11961:
11956:
11951:
11946:
11940:
11938:
11934:
11933:
11931:
11930:
11925:
11920:
11915:
11910:
11908:Function space
11905:
11900:
11895:
11889:
11887:
11883:
11882:
11880:
11879:
11874:
11873:
11872:
11862:
11857:
11851:
11849:
11834:
11830:
11817:
11816:
11814:
11813:
11808:
11803:
11798:
11793:
11788:
11783:
11781:CauchyâSchwarz
11778:
11772:
11770:
11761:
11760:
11758:
11757:
11752:
11747:
11741:
11739:
11730:
11729:
11727:
11726:
11721:
11716:
11707:
11702:
11701:
11700:
11690:
11682:
11680:Hilbert spaces
11672:
11670:
11669:Basic concepts
11666:
11665:
11658:
11657:
11650:
11643:
11635:
11607:
11606:
11594:
11585:
11579:
11566:
11560:
11547:
11541:
11525:
11519:
11506:
11500:
11485:
11471:
11454:
11448:
11435:
11429:
11416:
11410:
11389:
11383:
11370:
11364:
11343:
11334:
11328:
11314:
11312:
11309:
11306:
11305:
11281:
11279:, Theorem 14.5
11269:
11257:
11245:
11224:
11206:
11183:
11171:
11159:
11147:
11132:
11130:, Example 5.26
11120:
11108:
11095:
11094:
11092:
11089:
11088:
11087:
11082:
11077:
11072:
11067:
11062:
11057:
11047:
11042:
11037:
11030:
11027:
11026:
11025:
11024:
11023:
11007:
11002:
10997:
10990:
10986:
10981:
10977:
10967:
10955:
10952:
10949:
10944:
10939:
10934:
10930:
10926:
10922:
10918:
10915:
10910:
10905:
10898:
10894:
10889:
10885:
10881:
10870:
10858:
10845:
10841:
10835:
10825:
10821:
10798:
10788:
10783:
10774:
10770:
10758:
10749:
10735:
10727:
10722:
10715:
10711:
10692:
10691:
10682:
10680:
10669:
10662:
10658:
10653:
10647:
10643:
10638:
10633:
10626:
10623:
10620:
10617:
10614:
10610:
10606:
10605:
10602:
10599:
10595:
10589:
10585:
10580:
10576:
10575:
10572:
10560:
10554:
10550:
10546:
10545:
10542:
10529:
10528:
10517:
10516:
10515:
10510:
10503:
10498:
10497:
10496:
10482:
10478:
10474:
10471:
10468:
10463:
10459:
10455:
10450:
10446:
10424:
10419:
10418:
10417:
10404:
10386:
10371:
10353:
10348:
10347:
10346:
10335:
10332:
10329:
10325:
10321:
10318:
10314:
10310:
10307:
10302:
10298:
10293:
10288:
10283:
10277:
10273:
10253:
10252:almost surely.
10249:
10241:
10227:
10220:
10209:
10199:
10191:
10183:
10170:
10163:
10152:
10130:
10108:
10091:
10086:
10085:
10084:
10073:
10070:
10067:
10064:
10061:
10050:
10045:
10037:
10032:
10028:
10024:
10019:
10015:
10011:
10007:
10000:
9989:
9984:
9974:
9970:
9960:
9957:
9946:
9941:
9931:
9927:
9896:
9887:
9873:
9862:
9857:
9856:
9855:
9841:
9838:
9835:
9832:
9829:
9811:
9802:
9788:
9783:
9781:is a constant.
9765:
9762:
9759:
9756:
9753:
9742:
9737:
9729:
9724:
9720:
9716:
9711:
9707:
9703:
9699:
9692:
9681:
9676:
9666:
9662:
9652:
9649:
9638:
9633:
9623:
9619:
9594:
9591:
9588:
9585:
9582:
9564:
9555:
9541:
9530:
9525:
9524:
9523:
9512:
9501:
9496:
9486:
9482:
9477:
9470:
9459:
9454:
9445:
9439:
9435:
9431:
9426:
9422:
9417:
9407:
9404:
9393:
9388:
9378:
9374:
9354:
9347:
9340:
9329:
9324:
9323:
9322:
9320:is a constant.
9304:
9301:
9290:
9285:
9275:
9271:
9266:
9262:
9251:
9246:
9236:
9232:
9211:
9198:
9192:
9191:
9190:
9168:
9165:
9152:
9148:
9142:
9132:
9128:
9123:
9119:
9106:
9102:
9096:
9086:
9082:
9064:
9063:
9062:
9051:
9040:
9035:
9025:
9021:
9016:
9012:
8999:
8995:
8989:
8979:
8975:
8957:
8956:
8955:
8944:
8933:
8928:
8918:
8914:
8909:
8905:
8894:
8889:
8879:
8875:
8861:
8860:
8859:
8848:
8836:
8825:
8821:
8816:
8811:
8807:
8797:
8792:
8783:
8779:
8756:
8751:
8747:
8743:
8732:
8731:
8730:
8719:
8708:
8703:
8693:
8689:
8684:
8680:
8667:
8657:
8653:
8635:
8634:
8616:
8612:
8607:
8605:
8602:
8597:
8593:
8588:
8586:
8583:
8574:
8569:
8568:
8563:
8560:
8557:
8545:
8541:
8535:
8530:
8525:
8522:
8519:
8516:
8513:
8509:
8504:
8495:
8491:
8485:
8480:
8479:
8465:
8464:
8461:
8441:
8438:
8435:
8432:
8429:
8416:
8412:
8406:
8396:
8392:
8388:
8385:
8380:
8376:
8372:
8352:
8339:
8335:
8329:
8319:
8315:
8294:
8281:
8277:
8271:
8261:
8257:
8245:
8233:
8230:
8214:
8207:
8203:
8197:
8193:
8164:
8161:
8158:
8155:
8152:
8136:
8126:
8122:
8118:
8115:
8110:
8106:
8102:
8082:
8066:
8056:
8052:
8031:
8015:
8005:
8001:
7989:
7977:
7974:
7963:
7958:
7951:
7947:
7941:
7937:
7908:
7905:
7902:
7899:
7896:
7885:
7880:
7870:
7866:
7862:
7859:
7854:
7850:
7846:
7826:
7815:
7810:
7800:
7796:
7775:
7764:
7759:
7749:
7745:
7733:
7732:almost surely.
7721:
7718:
7715:
7695:
7682:
7678:
7672:
7662:
7658:
7637:
7624:
7620:
7614:
7604:
7600:
7588:
7587:almost surely.
7576:
7573:
7570:
7550:
7534:
7524:
7520:
7499:
7483:
7473:
7469:
7457:
7442:
7439:
7436:
7416:
7405:
7400:
7390:
7386:
7365:
7354:
7349:
7339:
7335:
7314:
7311:
7290:
7284:
7280:
7273:
7269:
7254:
7253:
7242:
7239:
7234:
7229:
7224:
7220:
7216:
7212:
7208:
7205:
7200:
7195:
7188:
7184:
7179:
7175:
7171:
7165:
7162:
7159:
7155:
7150:
7142:
7135:
7131:
7126:
7120:
7116:
7073:-th mean, for
7067:
7066:
7049:
7030:
7025:
7012:
6993:
6986:-th mean are:
6978:
6977:
6968:
6966:
6951:
6948:
6939:
6935:
6930:
6922:
6918:
6882:
6860:
6856:
6836:expected value
6832:
6831:
6820:
6817:
6814:
6810:
6804:
6799:
6794:
6791:
6786:
6782:
6777:
6772:
6767:
6761:
6758:
6755:
6751:
6730:
6711:
6697:
6682:
6639:
6626:
6623:
6610:
6590:
6570:
6550:
6545:
6541:
6537:
6517:
6514:
6511:
6508:
6505:
6502:
6497:
6493:
6489:
6484:
6480:
6479:lim sup
6476:
6473:
6449:
6446:
6443:
6438:
6434:
6430:
6410:
6407:
6404:
6401:
6398:
6393:
6389:
6385:
6382:
6377:
6374:
6371:
6367:
6343:
6340:
6335:
6331:
6310:
6288:
6285:
6280:
6277:
6274:
6271:
6267:
6261:
6257:
6252:
6248:
6245:
6225:
6221:
6217:
6214:
6211:
6208:
6205:
6183:
6180:
6175:
6172:
6169:
6166:
6163:
6160:
6155:
6151:
6147:
6144:
6122:
6119:
6114:
6111:
6108:
6105:
6100:
6096:
6092:
6089:
6069:
6064:
6060:
6056:
6044:
6041:
6021:
6018:
6015:
6011:
6007:
6004:
6001:
5998:
5995:
5992:
5989:
5986:
5981:
5977:
5971:
5968:
5965:
5961:
5957:
5954:
5951:
5948:
5944:
5900:
5897:
5894:
5891:
5888:
5885:
5882:
5879:
5876:
5871:
5867:
5861:
5858:
5855:
5851:
5844:
5841:
5838:
5835:
5832:
5803:random process
5792:
5781:
5778:
5777:
5776:
5768:
5755:
5752:
5740:
5737:
5732:
5727:
5720:
5717:
5714:
5710:
5702:
5697:
5694:
5691:
5688:
5685:
5682:
5679:
5676:
5671:
5667:
5661:
5656:
5652:
5649:
5646:
5643:
5638:
5632:
5611:
5608:
5605:
5602:
5599:
5583:
5572:
5571:
5562:
5560:
5545:
5542:
5534:
5531:
5528:
5525:
5520:
5512:
5508:
5473:
5470:
5467:
5457:
5454:
5449:
5442:
5437:
5434:
5430:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5400:
5396:
5391:
5387:
5384:
5381:
5378:
5373:
5366:
5363:
5360:
5356:
5355:lim sup
5350:
5344:
5317:
5314:
5309:
5304:
5301:
5298:
5295:
5292:
5289:
5286:
5283:
5278:
5274:
5268:
5265:
5262:
5258:
5254:
5251:
5248:
5245:
5240:
5234:
5209:
5205:
5201:
5196:
5191:
5188:
5185:
5162:
5151:
5135:
5132:
5128:
5124:
5121:
5116:
5112:
5105:
5102:
5099:
5095:
5090:
5084:
5050:
5044:
5041:
5027:
5026:
5008:
5001:
4991:
4990:
4986:
4985:
4977:
4976:
4966:
4963:
4950:
4928:
4925:
4922:
4918:
4914:
4909:
4905:
4901:
4898:
4895:
4892:
4889:
4886:
4882:
4878:
4874:
4868:
4864:
4859:
4855:
4852:
4849:
4846:
4843:
4840:
4836:
4830:
4826:
4822:
4817:
4813:
4808:
4804:
4801:
4781:
4759:
4755:
4732:
4728:
4705:
4701:
4695:
4691:
4687:
4684:
4681:
4678:
4673:
4669:
4646:
4642:
4628:
4625:
4624:
4623:
4612:
4608:
4604:
4601:
4598:
4594:
4590:
4587:
4584:
4580:
4576:
4573:
4569:
4564:
4560:
4557:
4554:
4551:
4548:
4545:
4542:
4520:
4515:
4512:
4507:
4502:
4499:
4495:
4491:
4488:
4485:
4481:
4475:
4469:
4462:
4459:
4456:
4453:
4448:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4421:
4395:
4382:
4379:
4376:
4373:
4368:
4364:
4360:
4355:
4351:
4347:
4344:
4323:
4318:
4314:
4308:
4304:
4283:
4268:
4265:
4264:is a constant.
4258:
4253:
4250:
4249:
4248:
4237:
4234:
4229:
4224:
4221:
4218:
4215:
4212:
4207:
4203:
4199:
4196:
4191:
4185:
4181:
4178:
4175:
4172:
4169:
4137:
4129:
4128:
4119:
4117:
4106:
4103:
4100:
4095:
4091:
4083:
4080:
4077:
4073:
4062:
4059:
4051:
4047:
4038:
4034:
4024:
4021:
4013:
4009:
4000:
3996:
3961:
3933:
3890:
3874:
3861:
3855:
3854:
3843:
3840:
3835:
3830:
3827:
3823:
3819:
3816:
3811:
3807:
3802:
3796:
3790:
3784:
3781:
3778:
3774:
3744:
3737:
3734:
3717:
3716:
3707:
3696:
3685:
3674:
3673:
3669:
3668:
3659:
3640:
3626:
3625:
3615:
3612:
3611:
3610:
3603:
3592:
3574:
3556:
3546:
3545:
3544:
3533:
3529:
3508:
3501:
3482:
3467:
3442:
3423:
3408:
3407:
3406:
3405:bounded below.
3394:
3370:
3367:
3364:
3361:
3357:
3353:
3350:
3345:
3341:
3337:
3334:
3330:
3326:
3325:lim inf
3316:
3315:bounded above;
3304:
3280:
3277:
3274:
3271:
3267:
3263:
3260:
3255:
3251:
3247:
3244:
3240:
3236:
3235:lim sup
3226:
3214:
3194:
3171:
3168:
3165:
3162:
3159:
3155:
3151:
3148:
3145:
3142:
3137:
3133:
3129:
3125:
3114:
3102:
3079:
3076:
3073:
3070:
3067:
3063:
3059:
3056:
3053:
3050:
3045:
3041:
3037:
3033:
3029:
3026:
3016:
3004:
2981:
2978:
2975:
2972:
2969:
2965:
2961:
2958:
2955:
2952:
2947:
2943:
2939:
2935:
2931:
2928:
2918:
2906:
2886:
2883:
2880:
2877:
2873:
2869:
2866:
2861:
2857:
2853:
2850:
2846:
2842:
2839:
2829:
2817:
2794:
2791:
2788:
2785:
2781:
2777:
2774:
2769:
2765:
2761:
2758:
2754:
2743:
2740:expected value
2726:
2705:
2678:
2675:
2672:
2669:
2665:
2661:
2658:
2653:
2649:
2645:
2642:
2638:
2627:
2615:
2612:
2609:
2606:
2603:
2599:
2595:
2592:
2572:
2569:
2566:
2563:
2560:
2556:
2552:
2549:
2546:
2543:
2538:
2534:
2530:
2526:
2504:
2493:
2492:
2491:
2471:
2463:) = (1 + cos(2
2455:
2445:
2428:
2414:
2411:
2408:
2405:
2402:
2398:
2394:
2391:
2388:
2385:
2382:
2369:
2366:
2358:
2336:
2335:
2324:
2321:
2318:
2315:
2310:
2306:
2303:
2298:
2294:
2290:
2287:
2282:
2277:
2254:
2240:
2203:continuity set
2188:
2183:
2178:
2175:
2164:
2163:
2152:
2149:
2146:
2143:
2140:
2136:
2132:
2129:
2126:
2123:
2118:
2114:
2110:
2106:
2100:
2097:
2094:
2090:
2052:
2047:
2042:
2038:
2034:
2031:
2026:
2022:
2018:
2013:
2009:
2004:
1993:random vectors
1977:
1974:
1970:
1967:
1964:
1959:
1950:
1946:
1938:
1934:
1902:
1896:
1879:
1878:
1869:
1867:
1852:
1849:
1846:
1843:
1838:
1833:
1830:
1825:
1821:
1817:
1812:
1801:
1798:
1795:
1790:
1786:
1776:
1773:
1770:
1765:
1761:
1757:
1755:
1752:
1747:
1741:
1730:
1726:
1717:
1713:
1703:
1700:
1691:
1686:
1677:
1673:
1663:
1660:
1651:
1646:
1637:
1633:
1623:
1620:
1612:
1608:
1599:
1595:
1591:
1587:
1586:
1558:
1538:
1535:
1532:
1512:
1492:
1489:
1486:
1483:
1480:
1475:
1471:
1461:, even though
1450:
1447:
1444:
1441:
1438:
1435:
1415:
1412:
1409:
1387:
1384:
1379:
1376:
1356:
1353:
1350:
1347:
1344:
1339:
1335:
1314:
1311:
1308:
1288:
1266:
1263:
1260:
1257:
1254:
1249:
1245:
1224:
1221:
1218:
1193:
1187:
1184:
1179:
1176:
1172:
1145:
1141:
1120:
1093:
1072:
1068:
1065:
1054:
1053:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1013:
1009:
1003:
1000:
997:
993:
946:
943:
938:
934:
930:
925:
921:
893:
890:
885:
881:
877:
872:
868:
855:
852:
834:
833:
794:
773:
769:
763:
758:
755:
752:
748:
739:
735:
728:
723:
719:
688:
680:
679:
675:
674:
653:
650:
647:
642:
638:
634:
629:
625:
619:
614:
610:
594:
570:
563:
536:
523:
517:
516:
512:
511:
499:
498:
488:
485:
472:
468:
464:
459:
454:
451:
448:
425:
405:
400:
396:
392:
359:
355:
334:
319:in probability
304:
300:
279:
268:
267:
256:
250:
246:
240:
235:
232:
229:
225:
219:
216:
211:
206:
202:
189:, is given by
170:
167:
164:
161:
158:
155:
152:
146:
141:
137:
111:
110:
103:
100:expected value
92:
91:
88:
85:
82:
79:
68:
65:
15:
9:
6:
4:
3:
2:
12175:
12164:
12161:
12159:
12156:
12155:
12153:
12136:
12133:
12131:
12128:
12127:
12126:
12123:
12121:
12120:Sobolev space
12118:
12116:
12115:Real analysis
12113:
12111:
12108:
12106:
12103:
12101:
12100:Lorentz space
12098:
12096:
12093:
12091:
12090:Bochner space
12088:
12087:
12085:
12081:
12071:
12068:
12066:
12063:
12061:
12058:
12054:
12051:
12050:
12049:
12046:
12044:
12041:
12040:
12038:
12035:
12029:
12023:
12020:
12018:
12015:
12013:
12010:
12008:
12005:
12003:
12000:
11999:
11997:
11995:
11991:
11985:
11982:
11980:
11977:
11975:
11972:
11970:
11967:
11965:
11962:
11960:
11957:
11955:
11952:
11950:
11947:
11945:
11942:
11941:
11939:
11935:
11929:
11926:
11924:
11921:
11919:
11916:
11914:
11911:
11909:
11906:
11904:
11901:
11899:
11896:
11894:
11891:
11890:
11888:
11884:
11878:
11875:
11871:
11868:
11867:
11866:
11863:
11861:
11858:
11856:
11853:
11852:
11850:
11848:
11828:
11818:
11812:
11809:
11807:
11804:
11802:
11799:
11797:
11794:
11792:
11791:Hilbert space
11789:
11787:
11784:
11782:
11779:
11777:
11774:
11773:
11771:
11769:
11767:
11762:
11756:
11753:
11751:
11748:
11746:
11743:
11742:
11740:
11738:
11736:
11731:
11725:
11722:
11720:
11717:
11715:
11711:
11708:
11706:
11705:Measure space
11703:
11699:
11696:
11695:
11694:
11691:
11689:
11687:
11683:
11681:
11677:
11674:
11673:
11671:
11667:
11663:
11656:
11651:
11649:
11644:
11642:
11637:
11636:
11633:
11629:
11628:
11626:
11622:
11618:
11614:
11600:
11595:
11591:
11586:
11582:
11576:
11572:
11567:
11563:
11557:
11553:
11548:
11544:
11538:
11534:
11530:
11526:
11522:
11516:
11512:
11507:
11503:
11497:
11493:
11492:
11486:
11482:
11478:
11474:
11468:
11464:
11460:
11455:
11451:
11445:
11441:
11436:
11432:
11426:
11422:
11417:
11413:
11407:
11403:
11399:
11395:
11390:
11386:
11380:
11376:
11371:
11367:
11361:
11357:
11352:
11351:
11344:
11340:
11335:
11331:
11325:
11321:
11316:
11315:
11295:
11291:
11285:
11278:
11273:
11266:
11261:
11255:, p. 354
11254:
11249:
11241:
11235:
11227:
11221:
11217:
11210:
11204:, Theorem 2.7
11203:
11198:
11196:
11194:
11192:
11190:
11188:
11181:, p. 289
11180:
11175:
11168:
11163:
11156:
11151:
11144:. p. 84.
11143:
11136:
11129:
11124:
11117:
11112:
11105:
11100:
11096:
11086:
11083:
11081:
11078:
11076:
11073:
11071:
11068:
11066:
11063:
11061:
11058:
11055:
11051:
11048:
11046:
11043:
11041:
11038:
11036:
11033:
11032:
11021:
11000:
10988:
10984:
10968:
10950:
10942:
10932:
10908:
10896:
10892:
10871:
10856:
10843:
10839:
10833:
10823:
10819:
10811:
10810:
10796:
10786:
10781:
10772:
10768:
10759:
10756:
10752:
10733:
10725:
10720:
10713:
10709:
10700:
10696:
10695:
10690:
10683:
10681:
10667:
10660:
10656:
10651:
10645:
10641:
10631:
10621:
10615:
10600:
10597:
10587:
10583:
10570:
10558:
10552:
10548:
10533:
10532:
10527:-convergence:
10526:
10522:
10518:
10499:
10480:
10476:
10472:
10469:
10466:
10461:
10457:
10453:
10448:
10444:
10436:
10435:
10434:
10433:
10431:
10420:
10415:
10411:
10398:
10393:
10382:
10378:
10367:
10363:
10359:
10349:
10333:
10327:
10323:
10319:
10316:
10308:
10305:
10300:
10296:
10286:
10275:
10271:
10263:
10262:
10261:
10260:
10258:
10254:
10248:
10245:converges to
10244:
10235:
10223:
10216:
10212:
10205:
10198:
10194:
10187:
10184:
10181:
10177:
10174:converges to
10159:
10146:
10141:
10137:
10134:converges in
10126:
10122:
10116:
10111:
10103:
10098:
10087:
10068:
10065:
10062:
10048:
10043:
10030:
10026:
10022:
10017:
10013:
9998:
9987:
9982:
9972:
9968:
9958:
9955:
9944:
9939:
9929:
9925:
9917:
9916:
9912:
9908:
9899:
9895:
9890:
9886:
9880:
9876:
9869:
9865:
9858:
9836:
9833:
9830:
9818:converges to
9814:
9810:
9805:
9801:
9795:
9784:
9780:
9760:
9757:
9754:
9740:
9735:
9722:
9718:
9714:
9709:
9705:
9690:
9679:
9674:
9664:
9660:
9650:
9647:
9636:
9631:
9621:
9617:
9609:
9608:
9589:
9586:
9583:
9567:
9563:
9558:
9554:
9548:
9544:
9537:
9526:
9510:
9499:
9494:
9484:
9480:
9468:
9457:
9452:
9437:
9433:
9429:
9424:
9420:
9405:
9402:
9391:
9386:
9376:
9372:
9364:
9363:
9361:
9357:
9350:
9343:
9336:
9325:
9319:
9302:
9299:
9288:
9283:
9273:
9269:
9260:
9249:
9244:
9234:
9230:
9222:
9221:
9219:
9214:
9210:
9206:
9201:
9197:
9193:
9187:
9183:
9166:
9163:
9150:
9146:
9140:
9130:
9126:
9117:
9104:
9100:
9094:
9084:
9080:
9072:
9071:
9069:
9065:
9049:
9038:
9033:
9023:
9019:
9010:
8997:
8993:
8987:
8977:
8973:
8965:
8964:
8962:
8958:
8942:
8931:
8926:
8916:
8912:
8903:
8892:
8887:
8877:
8873:
8865:
8864:
8862:
8846:
8834:
8823:
8819:
8814:
8805:
8795:
8790:
8781:
8777:
8769:
8768:
8749:
8745:
8733:
8717:
8706:
8701:
8691:
8687:
8678:
8665:
8655:
8651:
8643:
8642:
8640:
8639:
8638:
8614:
8610:
8595:
8591:
8572:
8543:
8539:
8533:
8523:
8520:
8517:
8514:
8511:
8493:
8489:
8483:
8470:
8469:
8468:
8462:
8439:
8436:
8433:
8430:
8427:
8414:
8410:
8404:
8394:
8390:
8386:
8383:
8378:
8374:
8370:
8350:
8337:
8333:
8327:
8317:
8313:
8292:
8279:
8275:
8269:
8259:
8255:
8246:
8231:
8228:
8212:
8205:
8201:
8195:
8191:
8162:
8159:
8156:
8153:
8150:
8134:
8124:
8120:
8116:
8113:
8108:
8104:
8100:
8080:
8064:
8054:
8050:
8029:
8013:
8003:
7999:
7990:
7975:
7972:
7961:
7956:
7949:
7945:
7939:
7935:
7906:
7903:
7900:
7897:
7894:
7883:
7878:
7868:
7864:
7860:
7857:
7852:
7848:
7844:
7824:
7813:
7808:
7798:
7794:
7773:
7762:
7757:
7747:
7743:
7734:
7719:
7716:
7713:
7693:
7680:
7676:
7670:
7660:
7656:
7635:
7622:
7618:
7612:
7602:
7598:
7589:
7574:
7571:
7568:
7548:
7532:
7522:
7518:
7497:
7481:
7471:
7467:
7458:
7455:
7454:almost surely
7440:
7437:
7434:
7414:
7403:
7398:
7388:
7384:
7363:
7352:
7347:
7337:
7333:
7324:
7323:
7322:
7320:
7310:
7308:
7288:
7282:
7278:
7271:
7267:
7240:
7232:
7222:
7206:
7198:
7186:
7182:
7157:
7140:
7133:
7129:
7124:
7118:
7114:
7103:
7102:
7101:
7098:
7096:
7092:
7088:
7084:
7080:
7076:
7072:
7064:
7060:
7056:
7045:
7041:
7037:
7034:converges in
7026:
7023:
7019:
7008:
7004:
7000:
6997:converges in
6989:
6988:
6987:
6985:
6976:
6969:
6967:
6949:
6946:
6937:
6933:
6928:
6920:
6916:
6905:
6904:
6901:
6899:
6894:
6880:
6858:
6854:
6837:
6818:
6815:
6812:
6808:
6802:
6792:
6789:
6784:
6780:
6770:
6753:
6741:
6740:
6739:
6737:
6726:
6700:
6672:
6664:
6660:
6659:
6657:
6650:
6648:
6633:
6622:
6608:
6588:
6568:
6543:
6539:
6515:
6512:
6503:
6500:
6495:
6491:
6482:
6471:
6464:ensures that
6463:
6444:
6441:
6436:
6432:
6399:
6396:
6391:
6387:
6380:
6375:
6372:
6369:
6365:
6355:
6341:
6333:
6329:
6308:
6286:
6283:
6278:
6272:
6269:
6259:
6255:
6243:
6223:
6219:
6215:
6212:
6209:
6206:
6203:
6181:
6178:
6173:
6170:
6167:
6161:
6158:
6153:
6149:
6142:
6120:
6117:
6112:
6106:
6103:
6098:
6094:
6087:
6062:
6058:
6040:
6037:
6032:
6019:
6013:
6009:
6002:
5996:
5993:
5987:
5979:
5975:
5963:
5955:
5949:
5946:
5942:
5933:
5931:
5927:
5922:
5920:
5916:
5911:
5898:
5892:
5886:
5883:
5877:
5869:
5865:
5853:
5842:
5836:
5833:
5822:
5820:
5816:
5812:
5808:
5804:
5800:
5795:
5791:
5787:
5773:
5769:
5766:
5762:
5761:Fatou's lemma
5758:
5757:
5751:
5738:
5735:
5725:
5712:
5692:
5686:
5683:
5677:
5669:
5665:
5654:
5650:
5644:
5641:
5606:
5603:
5600:
5590:
5586:
5579:
5570:
5563:
5561:
5543:
5540:
5532:
5526:
5518:
5510:
5506:
5495:
5494:
5491:
5489:
5484:
5471:
5468:
5465:
5455:
5452:
5435:
5432:
5421:
5415:
5412:
5406:
5398:
5394:
5385:
5379:
5376:
5358:
5333:
5328:
5315:
5312:
5299:
5293:
5290:
5284:
5276:
5272:
5260:
5252:
5246:
5243:
5223:
5199:
5189:
5175:
5174:
5173:Almost surely
5169:
5158:
5146:
5133:
5130:
5126:
5122:
5119:
5114:
5110:
5097:
5088:
5073:
5069:
5065:
5061:
5057:
5056:almost surely
5040:
5038:
5037:real analysis
5034:
5024:
5017:
5007:
5000:
4992:
4987:
4983:
4982:quite certain
4978:
4973:
4962:
4948:
4939:
4923:
4920:
4907:
4899:
4896:
4890:
4887:
4876:
4866:
4862:
4850:
4847:
4841:
4838:
4828:
4824:
4820:
4815:
4811:
4799:
4779:
4757:
4753:
4730:
4726:
4703:
4699:
4693:
4685:
4682:
4676:
4671:
4667:
4644:
4640:
4610:
4606:
4599:
4596:
4588:
4585:
4582:
4567:
4558:
4552:
4549:
4546:
4540:
4513:
4510:
4500:
4497:
4489:
4486:
4483:
4460:
4457:
4454:
4451:
4437:
4431:
4428:
4425:
4419:
4411:
4409:
4404:
4400:
4396:
4377:
4371:
4366:
4362:
4353:
4349:
4342:
4321:
4316:
4312:
4306:
4302:
4281:
4273:
4269:
4266:
4263:
4259:
4256:
4255:
4235:
4222:
4219:
4213:
4210:
4205:
4201:
4194:
4179:
4176:
4173:
4170:
4160:
4159:
4158:
4154:
4150:
4145:
4140:
4136:
4127:
4120:
4118:
4104:
4101:
4098:
4093:
4089:
4075:
4071:
4060:
4057:
4049:
4045:
4036:
4032:
4022:
4019:
4011:
4007:
3998:
3994:
3986:
3985:
3982:
3980:
3975:
3973:
3969:
3964:
3960:
3956:
3952:
3947:
3945:
3941:
3936:
3932:
3928:
3925: â„
3924:
3920:
3916:
3912:
3908:
3902:
3897:
3886:
3882:
3877:
3873:
3869:
3864:
3860:
3841:
3838:
3828:
3825:
3817:
3814:
3809:
3805:
3776:
3764:
3763:
3762:
3760:
3756:
3752:
3747:
3743:
3733:
3731:
3727:
3722:
3675:
3670:
3655:
3651:
3636:
3627:
3622:
3608:
3604:
3601:
3597:
3593:
3582:
3577:
3569:
3559:
3550:
3547:
3536:
3522:
3518:
3511:
3504:
3495:
3485:
3470:
3461:
3460:
3456:
3452:
3445:
3438:
3426:
3413:
3409:
3392:
3384:
3365:
3359:
3351:
3343:
3339:
3332:
3317:
3302:
3294:
3275:
3269:
3261:
3253:
3249:
3242:
3227:
3212:
3192:
3185:
3166:
3163:
3160:
3143:
3140:
3135:
3131:
3115:
3100:
3093:
3074:
3071:
3068:
3057:
3051:
3048:
3043:
3039:
3017:
3002:
2995:
2976:
2973:
2970:
2959:
2953:
2950:
2945:
2941:
2919:
2904:
2881:
2875:
2867:
2859:
2855:
2848:
2830:
2815:
2808:
2789:
2783:
2767:
2763:
2756:
2744:
2741:
2703:
2696:
2692:
2673:
2667:
2651:
2647:
2640:
2628:
2610:
2607:
2604:
2590:
2567:
2564:
2561:
2544:
2541:
2536:
2532:
2516:
2515:
2507:
2498:
2494:
2489:
2485:
2481:
2480:
2478:
2470:
2466:
2462:
2458:
2450:
2446:
2443:
2409:
2406:
2403:
2392:
2386:
2380:
2372:
2371:
2365:
2361:
2354:
2345:
2319:
2313:
2296:
2292:
2285:
2280:
2266:
2265:
2264:
2261:
2257:
2243:
2234:
2230:
2225:
2223:
2219:
2218:metric spaces
2216:in arbitrary
2215:
2210:
2204:
2186:
2176:
2173:
2147:
2144:
2141:
2130:
2124:
2121:
2116:
2112:
2092:
2080:
2079:
2078:
2068:
2050:
2040:
2036:
2032:
2029:
2024:
2020:
2016:
2011:
2007:
2002:
1994:
1989:
1972:
1968:
1965:
1948:
1944:
1936:
1932:
1900:
1877:
1870:
1868:
1850:
1844:
1823:
1819:
1799:
1796:
1788:
1784:
1774:
1771:
1763:
1759:
1750:
1745:
1728:
1724:
1715:
1711:
1701:
1698:
1684:
1675:
1671:
1661:
1658:
1644:
1635:
1631:
1621:
1618:
1610:
1606:
1597:
1593:
1577:
1576:
1573:
1570:
1556:
1536:
1533:
1530:
1510:
1490:
1487:
1481:
1473:
1469:
1448:
1445:
1439:
1433:
1413:
1410:
1407:
1385:
1382:
1377:
1374:
1354:
1351:
1345:
1337:
1333:
1312:
1309:
1306:
1286:
1279:
1264:
1261:
1255:
1247:
1243:
1222:
1219:
1216:
1208:
1191:
1185:
1182:
1177:
1174:
1170:
1162:on intervals
1161:
1143:
1139:
1118:
1109:
1107:
1091:
1066:
1063:
1040:
1034:
1028:
1025:
1019:
1011:
1007:
995:
983:
982:
981:
976:
968:
964:
960:
957:, is said to
944:
941:
936:
932:
928:
923:
919:
911:
907:
891:
888:
883:
879:
875:
870:
866:
851:
849:
843:
841:
829:
802:
790:
771:
767:
761:
756:
753:
750:
746:
737:
733:
726:
721:
717:
705:
701:
697:
691:
681:
676:
672:
668:
648:
645:
640:
636:
627:
623:
617:
612:
608:
597:
589:
578:
569:
562:
555:
551:and variance
548:
543:
535:
518:
515:Tossing coins
513:
506:
500:
495:
484:
462:
452:
439:
423:
398:
394:
381:
379:
375:
357:
353:
332:
324:
320:
302:
298:
277:
254:
248:
244:
238:
233:
230:
227:
223:
217:
214:
209:
204:
200:
192:
191:
190:
188:
184:
168:
165:
162:
159:
156:
153:
150:
144:
139:
135:
126:
123:
118:
114:
108:
104:
101:
97:
96:
95:
89:
86:
83:
80:
77:
74:
73:
72:
64:
61:
57:
53:
49:
44:
42:
38:
34:
30:
26:
22:
11937:Inequalities
11877:Uniform norm
11765:
11734:
11685:
11610:
11609:
11589:
11570:
11551:
11532:
11510:
11490:
11462:
11439:
11420:
11393:
11374:
11349:
11338:
11319:
11297:. Retrieved
11293:
11284:
11272:
11260:
11248:
11215:
11209:
11174:
11162:
11150:
11141:
11135:
11123:
11111:
11099:
10747:
10698:
10684:
10524:
10429:
10428:is a sum of
10409:
10396:
10391:
10380:
10376:
10365:
10361:
10357:
10256:
10246:
10239:
10233:
10218:
10214:
10207:
10203:
10196:
10189:
10185:
10179:
10178:also in any
10175:
10157:
10144:
10139:
10135:
10124:
10120:
10114:
10106:
10101:
10096:
9910:
9906:
9897:
9893:
9888:
9884:
9878:
9871:
9867:
9860:
9812:
9808:
9803:
9799:
9793:
9778:
9565:
9561:
9556:
9552:
9546:
9539:
9535:
9359:
9352:
9345:
9338:
9334:
9317:
9217:
9212:
9208:
9204:
9199:
9195:
9185:
9181:
9067:
8960:
8636:
8466:
7316:
7255:
7099:
7094:
7090:
7086:
7082:
7074:
7070:
7068:
7062:
7058:
7054:
7043:
7039:
7038:-th mean to
7035:
7021:
7017:
7006:
7002:
7001:-th mean to
6998:
6983:
6981:
6970:
6897:
6895:
6833:
6735:
6724:
6695:
6662:
6655:
6652:
6646:
6644:
6631:
6628:
6356:
6046:
6033:
5934:
5923:
5915:sample space
5912:
5823:
5818:
5814:
5810:
5806:
5805:) converges
5793:
5789:
5783:
5775:convergence.
5589:metric space
5581:
5576:For generic
5575:
5564:
5487:
5485:
5329:
5221:
5171:
5167:
5156:
5147:
5071:
5067:
5063:
5059:
5055:
5046:
5030:
5022:
5015:
5005:
4998:
4981:
4940:
4630:
4406:
4261:
4152:
4148:
4138:
4134:
4132:
4121:
3978:
3976:
3971:
3967:
3962:
3958:
3954:
3950:
3948:
3943:
3939:
3934:
3930:
3926:
3922:
3918:
3914:
3910:
3906:
3900:
3895:
3884:
3880:
3875:
3871:
3867:
3862:
3858:
3856:
3758:
3754:
3750:
3745:
3741:
3740:A sequence {
3739:
3723:
3720:
3634:
3572:
3554:
3527:
3520:
3516:
3506:
3499:
3493:
3480:
3465:
3454:
3450:
3440:
3436:
3421:
2738:denotes the
2502:
2483:
2476:
2468:
2464:
2460:
2453:
2356:
2352:
2343:
2337:
2259:
2252:
2250:(denoted as
2238:
2228:
2226:
2211:
2165:
2069:to a random
2066:
1990:
1882:
1871:
1571:
1110:
1055:
966:
962:
958:
857:
844:
837:
800:
703:
698:sequence of
686:
666:
592:
567:
560:
553:
546:
533:
531:times. Then
497:Dice factory
382:
318:
269:
121:
119:
115:
112:
93:
70:
59:
45:
41:distribution
36:
32:
28:
27:, including
24:
18:
12135:Von Neumann
11949:Chebyshev's
11613:Citizendium
11179:Dudley 2002
11167:Dudley 2002
11157:, Lemma 2.2
11118:, p. 4
10255:If for all
10138:th mean to
6738:exist, and
5074:means that
5016:almost sure
3898:if for any
3757:if for all
2201:which is a
858:A sequence
125:independent
76:Convergence
56:mathematics
12152:Categories
12130:C*-algebra
11954:Clarkson's
11311:References
11299:2022-03-12
7313:Properties
7053:converges
7016:converges
6643:converges
5811:everywhere
5754:Properties
5054:converges
5043:Definition
5014:We may be
4403:metrizable
4252:Properties
3736:Definition
3726:consistent
3596:metrizable
3381:for every
3291:for every
3092:closed set
3090:for every
2992:for every
2742:operator);
2368:Properties
2166:for every
1235:. Indeed,
1207:degenerate
1106:continuous
854:Definition
588:bell curve
577:binomially
317:converges
67:Background
48:statistics
12125:*-algebra
12110:Quasinorm
11979:Minkowski
11870:Essential
11833:∞
11662:Lp spaces
11615:article "
11267:, Th.2.19
11234:cite book
10954:∞
10917:→
10637:⇒
10625:∞
10470:⋯
10331:∞
10320:ε
10306:−
10272:∑
10231:for each
10123:and some
10113:| â€
10099:, and if
10006:⇒
9777:provided
9698:⇒
9476:⇒
9430:−
9316:provided
9265:⇒
9180:provided
9122:⇒
9015:⇒
8908:⇒
8810:⇒
8683:⇒
8604:⇒
8585:⇒
8562:⇓
8521:≥
8508:⇒
7164:∞
7161:→
7149:⇒
6790:−
6760:∞
6757:→
6665:, if the
6409:∞
6406:→
6373:≥
6366:∑
6339:→
6273:ε
6270:≥
6210:ε
6174:−
6017:Ω
6003:ω
5988:ω
5970:∞
5967:→
5953:Ω
5950:∈
5947:ω
5893:ω
5878:ω
5860:∞
5857:→
5843::
5840:Ω
5837:∈
5834:ω
5831:∀
5815:pointwise
5801:(i.e., a
5719:∞
5716:→
5709:⟶
5693:ω
5678:ω
5651::
5648:Ω
5645:∈
5642:ω
5466:ε
5436:ε
5422:ω
5413:−
5407:ω
5383:Ω
5380:∈
5377:ω
5365:∞
5362:→
5300:ω
5285:ω
5267:∞
5264:→
5250:Ω
5247:∈
5244:ω
5187:Ω
5104:∞
5101:→
4989:Example 2
4975:Example 1
4924:ϵ
4921:≥
4897:−
4891:−
4877:⋅
4842:ϵ
4839:≥
4821:−
4683:−
4586:−
4514:ε
4511:≤
4501:ε
4487:−
4452:ε
4233:→
4223:ε
4220:≥
4171:ε
4168:∀
4082:∞
4079:→
3829:ε
3815:−
3783:∞
3780:→
3652:) by the
3385:function
3352:≥
3295:function
3262:≤
3164:∈
3150:→
3141:∈
3072:∈
3058:≤
3049:∈
2974:∈
2960:≥
2951:∈
2868:≥
2776:→
2660:→
2608:≤
2594:↦
2565:≤
2551:→
2542:≤
2407:≤
2305:→
2281:∗
2177:⊂
2145:∈
2122:∈
2099:∞
2096:→
2041:⊂
2033:…
1832:→
1794:⇒
1769:⇝
1378:≥
1310:≤
1160:uniformly
1084:at which
1067:∈
1002:∞
999:→
945:…
892:…
747:∑
649:μ
646:−
628:σ
450:Ω
333:μ
224:∑
163:…
11974:Markov's
11969:Hölder's
11959:Hanner's
11776:Bessel's
11714:function
11698:Lebesgue
11461:(1991).
11029:See also
10834:→
10782:→
10721:→
10652:→
10559:→
10401:), then
10364:towards
10259:> 0,
10182:th mean.
10142:for all
10119:for all
10044:→
9983:→
9940:→
9736:→
9675:→
9632:→
9495:→
9453:→
9387:→
9284:→
9245:→
9141:→
9095:→
9034:→
8988:→
8927:→
8888:→
8835:→
8791:→
8702:→
8666:→
8611:→
8592:→
8573:→
8534:→
8484:→
8405:→
8328:→
8270:→
8213:→
8135:→
8065:→
8014:→
7957:→
7879:→
7809:→
7758:→
7671:→
7613:→
7533:→
7482:→
7399:→
7348:→
7319:complete
7279:→
7125:→
7018:in mean
6929:→
6649:-th mean
6236:we have
5817:towards
5772:topology
5519:→
5070:towards
5068:strongly
4772:for all
4399:topology
4363:→
4313:→
4046:→
4008:→
3905:and any
3182:for all
2994:open set
2689:for all
2073:-vector
1945:→
1725:→
1685:→
1645:→
1607:→
1503:for all
1367:for all
683:Suppose
665:will be
540:has the
270:then as
187:variance
11994:Results
11693:Measure
11481:1102015
10368:. When
10127:, then
10104:(|
9909:,
9892:,
9852:
9820:
9807:,
9605:
9573:
9560:,
9207:, then
8363:, then
8093:, then
7837:, then
7706:, then
7561:, then
7427:, then
6701:|) and
6653:in the
6645:in the
5587:} on a
5461:for all
4792:, but:
4405:by the
4142:} on a
3887:. Then
3761:> 0
3635:ex ante
3598:by the
3433:, then
2716:(where
2691:bounded
1278:for all
908:, with
817:
805:
706:(â1, 1)
700:uniform
11847:spaces
11768:spaces
11737:spaces
11688:spaces
11676:Banach
11577:
11558:
11539:
11517:
11498:
11479:
11469:
11446:
11427:
11408:
11381:
11362:
11326:
11222:
10854:
10829:
10794:
10778:
10399:> 0
10238:, and
10057:
10039:
10002:
9996:
9978:
9965:
9962:
9953:
9935:
9749:
9731:
9694:
9688:
9670:
9657:
9654:
9645:
9627:
9508:
9490:
9472:
9466:
9448:
9412:
9409:
9400:
9382:
9297:
9279:
9258:
9240:
9161:
9136:
9115:
9090:
9047:
9029:
9008:
8983:
8940:
8922:
8901:
8883:
8844:
8831:
8803:
8787:
8715:
8697:
8676:
8661:
8425:
8400:
8348:
8323:
8290:
8265:
8226:
8183:) and
8148:
8130:
8078:
8060:
8027:
8009:
7970:
7927:) and
7892:
7874:
7822:
7804:
7771:
7753:
7691:
7666:
7633:
7608:
7546:
7528:
7495:
7477:
7412:
7394:
7361:
7343:
6727:|) of
6357:Since
6321:hence
6196:. For
5846:
5821:means
5807:surely
4464:
4410:metric
4408:Ky Fan
4067:
4064:
4055:
4042:
4029:
4026:
4017:
4004:
3903:> 0
3524:or of
2373:Since
1883:where
1806:
1803:
1781:
1778:
1734:
1721:
1708:
1705:
1696:
1681:
1668:
1665:
1656:
1641:
1628:
1625:
1616:
1603:
1549:where
1325:, and
694:is an
556:= 0.25
148:
35:, and
11602:(PDF)
11091:Notes
10753:) is
10500:then
10360:, or
10117:) = 1
7085:>
7061:) to
7027:When
6990:When
6658:-norm
4294:, if
2472:(0,1)
2263:) if
1400:when
1299:when
973:with
965:, or
961:, or
573:, ...
549:= 0.5
12032:For
11886:Maps
11625:GFDL
11575:ISBN
11556:ISBN
11537:ISBN
11515:ISBN
11496:ISBN
11467:ISBN
11444:ISBN
11425:ISBN
11406:ISBN
11379:ISBN
11360:ISBN
11356:1â28
11324:ISBN
11240:link
11220:ISBN
10951:<
10622:<
10598:<
10564:a.s.
10519:The
10328:<
10317:>
9870:and
9538:and
9344:and
9188:â„ 1.
8839:a.s.
8670:a.s.
8577:a.s.
8515:>
8456:and
8305:and
8218:a.s.
8179:and
8140:a.s.
8070:a.s.
8042:and
8019:a.s.
7923:and
7786:and
7648:and
7538:a.s.
7510:and
7487:a.s.
7376:and
7057:(or
7042:for
7005:for
6873:and
6734:and
6669:-th
6651:(or
6213:<
6207:<
6135:and
5488:a.s.
5469:>
5433:>
4997:Let
4498:>
4455:>
4270:The
4174:>
4072:plim
3957:and
3917:and
3826:>
3477:and
3410:The
2495:The
1991:For
1411:>
803:(0,
520:Let
323:mean
185:and
183:mean
50:and
11398:doi
11018:is
10760:If
10421:If
10236:â„ 0
10147:â„ 1
10088:If
9859:If
9527:If
9326:If
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8247:If
7991:If
7735:If
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7325:If
7309:).
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7020:to
6750:lim
6634:â„ 1
6621:).
5960:lim
5850:lim
5813:or
5809:or
5257:lim
5094:lim
5066:or
5062:or
5058:or
4572:min
4441:inf
3773:lim
3587:of
3539:to
3514:to
3494:not
3488:to
3473:to
3446:)}
3028:sup
3025:lim
2930:inf
2927:lim
2841:inf
2838:lim
2465:Ïnx
2440:is
2364:â.
2205:of
2089:lim
2077:if
1104:is
992:lim
980:if
696:iid
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58:as
19:In
12154::
11477:MR
11475:.
11404:.
11358:.
11292:.
11236:}}
11232:{{
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9220::
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8460:).
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5039:.
5004:,
4412::
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3842:0.
3732:.
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3512:}
3505:+
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2258:â
2244:}
2209:.
1988:.
1108:.
850:.
692:}
566:,
507:.
483:.
380:.
325:,
31:,
11829:L
11766:L
11735:L
11712:/
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11647:t
11640:v
11627:.
11604:.
11583:.
11564:.
11545:.
11523:.
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11433:.
11414:.
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11332:.
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11228:.
11022:.
11006:}
11001:r
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10948:]
10943:r
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10925:[
10921:E
10914:]
10909:r
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10897:n
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10888:|
10884:[
10880:E
10869:,
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10844:r
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10824:n
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10787:p
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10757:.
10750:n
10748:X
10734:X
10726:P
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10710:X
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10685:(
10668:X
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10632:}
10619:]
10616:Y
10613:[
10609:E
10601:Y
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10579:|
10571:X
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10549:X
10525:L
10511:n
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10504:n
10502:S
10481:n
10477:X
10473:+
10467:+
10462:1
10458:X
10454:=
10449:n
10445:S
10430:n
10425:n
10423:S
10416:.
10410:X
10405:n
10403:X
10397:Δ
10392:X
10387:n
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10381:X
10377:X
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10313:|
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10301:n
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10292:|
10287:(
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10257:Δ
10250:0
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10219:Y
10215:n
10210:n
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10204:F
10200:0
10197:X
10192:n
10190:X
10180:r
10176:X
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10151:X
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10140:X
10136:r
10131:n
10129:X
10125:b
10121:n
10115:b
10109:n
10107:X
10102:P
10097:X
10092:n
10090:X
10072:)
10069:Y
10066:,
10063:X
10060:(
10049:p
10036:)
10031:n
10027:Y
10023:,
10018:n
10014:X
10010:(
9999:Y
9988:p
9973:n
9969:Y
9959:,
9956:X
9945:p
9930:n
9926:X
9913:)
9911:Y
9907:X
9905:(
9901:)
9898:n
9894:Y
9889:n
9885:X
9883:(
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9874:n
9872:Y
9868:X
9863:n
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9854:.
9840:)
9837:Y
9834:,
9831:X
9828:(
9816:)
9813:n
9809:Y
9804:n
9800:X
9798:(
9794:Y
9789:n
9787:Y
9779:c
9764:)
9761:c
9758:,
9755:X
9752:(
9741:d
9728:)
9723:n
9719:Y
9715:,
9710:n
9706:X
9702:(
9691:c
9680:d
9665:n
9661:Y
9651:,
9648:X
9637:d
9622:n
9618:X
9593:)
9590:c
9587:,
9584:X
9581:(
9569:)
9566:n
9562:Y
9557:n
9553:X
9551:(
9547:c
9542:n
9540:Y
9536:X
9531:n
9529:X
9511:X
9500:d
9485:n
9481:Y
9469:0
9458:p
9444:|
9438:n
9434:Y
9425:n
9421:X
9416:|
9406:,
9403:X
9392:d
9377:n
9373:X
9360:X
9355:n
9353:Y
9348:n
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9341:n
9339:X
9335:X
9330:n
9328:X
9318:c
9303:,
9300:c
9289:p
9274:n
9270:X
9261:c
9250:d
9235:n
9231:X
9218:c
9213:n
9209:X
9205:c
9200:n
9196:X
9186:s
9182:r
9167:,
9164:X
9151:s
9147:L
9131:n
9127:X
9118:X
9105:r
9101:L
9085:n
9081:X
9068:r
9050:X
9039:p
9024:n
9020:X
9011:X
8998:r
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8978:n
8974:X
8961:r
8943:X
8932:d
8917:n
8913:X
8904:X
8893:p
8878:n
8874:X
8847:X
8824:k
8820:n
8815:X
8806:X
8796:p
8782:n
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8755:)
8750:k
8746:n
8742:(
8718:X
8707:p
8692:n
8688:X
8679:X
8656:n
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8615:d
8596:p
8544:r
8540:L
8524:1
8518:r
8512:s
8494:s
8490:L
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8454:a
8440:Y
8437:b
8434:+
8431:X
8428:a
8415:r
8411:L
8395:n
8391:Y
8387:b
8384:+
8379:n
8375:X
8371:a
8351:Y
8338:r
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8318:n
8314:Y
8293:X
8280:r
8276:L
8260:n
8256:X
8244:.
8232:Y
8229:X
8206:n
8202:Y
8196:n
8192:X
8181:b
8177:a
8163:Y
8160:b
8157:+
8154:X
8151:a
8125:n
8121:Y
8117:b
8114:+
8109:n
8105:X
8101:a
8081:Y
8055:n
8051:Y
8030:X
8004:n
8000:X
7988:.
7976:Y
7973:X
7962:p
7950:n
7946:Y
7940:n
7936:X
7925:b
7921:a
7907:Y
7904:b
7901:+
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7895:a
7884:p
7869:n
7865:Y
7861:b
7858:+
7853:n
7849:X
7845:a
7825:Y
7814:p
7799:n
7795:Y
7774:X
7763:p
7748:n
7744:X
7720:Y
7717:=
7714:X
7694:Y
7681:r
7677:L
7661:n
7657:X
7636:X
7623:r
7619:L
7603:n
7599:X
7575:Y
7572:=
7569:X
7549:Y
7523:n
7519:X
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7468:X
7456:.
7441:Y
7438:=
7435:X
7415:Y
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7364:X
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7289:X
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7241:.
7238:]
7233:r
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7219:|
7215:[
7211:E
7207:=
7204:]
7199:r
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7174:[
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6973:4
6971:(
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6813:=
6809:)
6803:r
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6793:X
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6766:E
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6549:}
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6540:X
6536:{
6516:1
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6507:}
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6501:=
6496:n
6492:X
6488:{
6483:n
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6472:P
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6445:1
6442:=
6437:n
6433:X
6429:{
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6392:n
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6330:X
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6266:|
6260:n
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6224:2
6220:/
6216:1
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6162:0
6159:=
6154:n
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6146:(
6143:P
6121:n
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6110:)
6107:1
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6099:n
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6088:P
6068:}
6063:n
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6055:{
6020:.
6014:=
6010:}
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6000:(
5997:X
5994:=
5991:)
5985:(
5980:n
5976:X
5964:n
5956::
5943:{
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5890:(
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5884:=
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5875:(
5870:n
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5767:.
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5736:=
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5675:(
5670:n
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5429:|
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5399:n
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5390:|
5386::
5372:{
5359:n
5349:(
5343:P
5313:=
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5294:X
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