2303:
theorem says that provided the finite-dimensional distributions satisfy the obvious consistency requirements, one can always identify a probability space to match the purpose. In many situations, this means that one does not have to be explicit about what the probability space is. Many texts on stochastic processes do, indeed, assume a probability space but never state explicitly what it is.
952:
700:
620:
2302:
The measure-theoretic approach to stochastic processes starts with a probability space and defines a stochastic process as a family of functions on this probability space. However, in many applications the starting point is really the finite-dimensional distributions of the stochastic process. The
1225:
2298:
Since the two conditions are trivially satisfied for any stochastic process, the power of the theorem is that no other conditions are required: For any reasonable (i.e., consistent) family of finite-dimensional distributions, there exists a stochastic process with these distributions.
2136:
947:{\displaystyle \nu _{t_{1}\dots t_{k}}\left(F_{1}\times \dots \times F_{k}\right)=\nu _{t_{1}\dots t_{k},t_{k+1},\dots ,t_{k+m}}\left(F_{1}\times \dots \times F_{k}\times \underbrace {\mathbb {R} ^{n}\times \dots \times \mathbb {R} ^{n}} _{m}\right).}
428:
2288:
1057:
2018:
1952:
3547:
2478:
2741:
2647:
1049:
1002:
3060:
1886:
2388:
defined on the finite products of these spaces would suffice, provided that these measures satisfy a certain compatibility relation. The formal statement of the general theorem is as follows.
3292:
1331:
665:
420:
3699:
2023:
1489:
3423:
2779:
3118:
1560:
3772:
1669:
1398:
314:
265:
218:
1795:
1622:
3605:
3651:
2989:
2683:
1289:
693:
162:
134:
2382:
2353:
1709:
1438:
615:{\displaystyle \nu _{t_{\pi (1)}\dots t_{\pi (k)}}\left(F_{\pi (1)}\times \dots \times F_{\pi (k)}\right)=\nu _{t_{1}\dots t_{k}}\left(F_{1}\times \dots \times F_{k}\right);}
378:
3354:
3181:
2957:
3231:
2924:
2862:
2562:
3318:
2808:
2588:
1261:
2897:
2531:
1562:. Therefore, an alternative way of stating Kolmogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure
3149:
2835:
3725:
3449:
2504:
2310:, by specifying the finite dimensional distributions to be Gaussian random variables, satisfying the consistency conditions above. As in most of the definitions of
2190:
2163:
3378:
3204:
1805:
The two conditions required by the theorem are trivially satisfied by any stochastic process. For example, consider a real-valued discrete-time stochastic process
1749:
1580:
340:
2409:
1823:
1729:
1509:
1351:
98:
2198:
2326:
The
Kolmogorov extension theorem gives us conditions for a collection of measures on Euclidean spaces to be the finite-dimensional distributions of some
1220:{\displaystyle \nu _{t_{1}\dots t_{k}}\left(F_{1}\times \dots \times F_{k}\right)=\mathbb {P} \left(X_{t_{1}}\in F_{1},\dots ,X_{t_{k}}\in F_{k}\right)}
3930:
3897:
J. Aldrich, But you have to remember PJ Daniell of
Sheffield, Electronic Journal for History of Probability and Statistics, Vol. 3, number 2, 2007
1957:
1891:
3454:
3785:
This theorem has many far-reaching consequences; for example it can be used to prove the existence of the following, among others:
2414:
3882:
2688:
2596:
1007:
963:
2997:
1828:
2290:. Also this is a trivial condition that will be satisfied by any consistent family of finite-dimensional distributions.
2131:{\displaystyle \nu _{1,2}(\mathbb {R} _{+}\times \mathbb {R} _{-})=\nu _{2,1}(\mathbb {R} _{-}\times \mathbb {R} _{+})}
3848:
3236:
1294:
628:
383:
3656:
1446:
3386:
3874:
2746:
51:
3068:
1514:
3380:
may sometimes be extended appropriately to a larger sigma algebra, if there is additional structure involved.
3730:
2315:
1627:
1356:
274:
223:
171:
1758:
1585:
3727:. The reason that the original statement of the theorem does not mention inner regularity of the measures
3552:
3610:
2962:
2656:
20:
19:
This article is about a theorem on stochastic processes. For a theorem on extension of pre-measure, see
2318:
to construct a continuous modification of the process constructed by the
Kolmogorov extension theorem.
1266:
670:
139:
111:
2358:
2329:
1674:
1403:
345:
3326:
3154:
2929:
3209:
2902:
2840:
2540:
3297:
2787:
2567:
1233:
2875:
2509:
101:
3868:
3127:
2813:
3911:
3838:
2385:
3704:
3428:
3383:
Note that the original statement of the theorem is just a special case of this theorem with
2483:
1752:
2384:
is unnecessary. In fact, any collection of measurable spaces together with a collection of
2168:
2141:
3363:
3189:
1734:
1565:
325:
8:
3121:
2314:
it is required that the sample paths are continuous almost surely, and one then uses the
268:
3817:
2394:
2283:{\displaystyle \mathbb {P} (X_{1}>0)=\mathbb {P} (X_{1}>0,X_{2}\in \mathbb {R} )}
2138:. The first condition generalizes this statement to hold for any number of time points
1808:
1714:
1494:
1336:
83:
59:
55:
3360:
on their respective spaces, which (as mentioned before) is rather coarse. The measure
2020:. Hence, for the finite-dimensional distributions to be consistent, it must hold that
1731:
is uncountable, but the price to pay for this level of generality is that the measure
3878:
3844:
958:
69:
2869:
16:
Consistent set of finite-dimensional distributions will define a stochastic process
3914:
2534:
2311:
2307:
3813:
3790:
2959:
of measures satisfies the following compatibility relation: for finite subsets
3924:
3779:
3774:
is that this would automatically follow, since Borel probability measures on
3357:
2865:
66:
3797:
3775:
1443:
In fact, it is always possible to take as the underlying probability space
3877:. Vol. 126. Providence: American Mathematical Society. p. 195.
3864:
320:
27:
3812:
According to John
Aldrich, the theorem was independently discovered by
2355:-valued stochastic process, but the assumption that the state space be
3840:
Stochastic
Differential Equations: An Introduction with Applications
3800:
taking values in a given state space with a given transition matrix,
2306:
The theorem is used in one of the standard proofs of existence of a
2013:{\displaystyle \nu _{2,1}(\mathbb {R} _{-}\times \mathbb {R} _{+})}
1947:{\displaystyle \nu _{1,2}(\mathbb {R} _{+}\times \mathbb {R} _{-})}
165:
3542:{\displaystyle \mu _{\{t_{1},...,t_{k}\}}=\nu _{t_{1}\dots t_{k}}}
47:
316:
Suppose that these measures satisfy two consistency conditions:
3607:. The stochastic process would simply be the canonical process
63:
2473:{\displaystyle \{(\Omega _{t},{\mathcal {F}}_{t})\}_{t\in T}}
3915:
Electronic Journ@l for
History of Probability and Statistics
105:
50:
that guarantees that a suitably "consistent" collection of
2195:
Continuing the example, the second condition implies that
1400:
as its finite-dimensional distributions relative to times
3820:
in the slightly different setting of integration theory.
3803:
infinite products of (inner-regular) probability spaces.
2736:{\displaystyle \pi _{I}^{J}:\Omega _{J}\to \Omega _{I}}
2642:{\displaystyle \Omega _{J}:=\prod _{t\in J}\Omega _{t}}
2480:
be some collection of measurable spaces, and for each
3733:
3707:
3659:
3613:
3555:
3457:
3431:
3389:
3366:
3329:
3300:
3239:
3212:
3192:
3157:
3130:
3071:
3000:
2965:
2932:
2905:
2878:
2843:
2816:
2790:
2749:
2691:
2659:
2599:
2570:
2543:
2512:
2486:
2417:
2397:
2361:
2332:
2201:
2171:
2144:
2026:
1960:
1894:
1831:
1811:
1761:
1737:
1717:
1677:
1630:
1588:
1568:
1517:
1497:
1449:
1406:
1359:
1339:
1297:
1269:
1236:
1060:
1044:{\displaystyle X:T\times \Omega \to \mathbb {R} ^{n}}
1010:
997:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
966:
703:
673:
631:
431:
386:
348:
328:
277:
226:
174:
142:
114:
86:
3912:"But you have to remember P.J.Daniell of Sheffield"
3055:{\displaystyle \mu _{F}=(\pi _{F}^{G})_{*}\mu _{G}}
1881:{\displaystyle \mathbb {P} (X_{1}>0,X_{2}<0)}
3766:
3719:
3693:
3645:
3599:
3541:
3443:
3417:
3372:
3348:
3312:
3286:
3225:
3198:
3175:
3143:
3112:
3054:
2983:
2951:
2918:
2891:
2856:
2829:
2802:
2773:
2735:
2677:
2641:
2582:
2556:
2525:
2498:
2472:
2403:
2376:
2347:
2282:
2184:
2157:
2130:
2012:
1946:
1880:
1817:
1789:
1743:
1723:
1703:
1663:
1616:
1574:
1554:
1503:
1483:
1432:
1392:
1345:
1325:
1283:
1255:
1219:
1043:
996:
946:
687:
659:
614:
414:
372:
334:
308:
259:
212:
156:
128:
92:
3922:
3843:(Sixth ed.). Berlin: Springer. p. 11.
1800:
3836:
3287:{\displaystyle \mu _{F}=(\pi _{F}^{T})_{*}\mu }
3186:Then there exists a unique probability measure
1326:{\displaystyle F_{i}\subseteq \mathbb {R} ^{n}}
660:{\displaystyle F_{i}\subseteq \mathbb {R} ^{n}}
415:{\displaystyle F_{i}\subseteq \mathbb {R} ^{n}}
3694:{\displaystyle \Omega =(\mathbb {R} ^{n})^{T}}
2321:
2293:
1711:. Kolmogorov's extension theorem applies when
1484:{\displaystyle \Omega =(\mathbb {R} ^{n})^{T}}
58:. It is credited to the English mathematician
3501:
3463:
3418:{\displaystyle \Omega _{t}=\mathbb {R} ^{n}}
2946:
2933:
2455:
2418:
367:
349:
2774:{\displaystyle \omega \mapsto \omega |_{I}}
75:
3113:{\displaystyle (\pi _{F}^{G})_{*}\mu _{G}}
1555:{\displaystyle X\colon (t,Y)\mapsto Y_{t}}
3671:
3405:
2364:
2335:
2273:
2233:
2203:
2115:
2100:
2063:
2048:
1997:
1982:
1931:
1916:
1833:
1767:
1594:
1461:
1313:
1277:
1134:
1031:
987:
914:
893:
681:
647:
402:
283:
150:
122:
3151:induced by the canonical projection map
2810:, suppose we have a probability measure
3931:Theorems regarding stochastic processes
3767:{\displaystyle \nu _{t_{1}\dots t_{k}}}
1664:{\displaystyle \nu _{t_{1}\dots t_{k}}}
1393:{\displaystyle \nu _{t_{1}\dots t_{k}}}
309:{\displaystyle (\mathbb {R} ^{n})^{k}.}
260:{\displaystyle \nu _{t_{1}\dots t_{k}}}
213:{\displaystyle t_{1},\dots ,t_{k}\in T}
3923:
1790:{\displaystyle (\mathbb {R} ^{n})^{T}}
1617:{\displaystyle (\mathbb {R} ^{n})^{T}}
3600:{\displaystyle t_{1},...,t_{k}\in T}
2926:. Suppose also that this collection
2743:denote the canonical projection map
3863:
3646:{\displaystyle (\pi _{t})_{t\in T}}
2984:{\displaystyle F\subset G\subset T}
2678:{\displaystyle I\subset J\subset T}
1671:for any finite collection of times
13:
3660:
3391:
3214:
2907:
2845:
2724:
2711:
2630:
2601:
2545:
2440:
2425:
1450:
1023:
978:
970:
14:
3942:
3904:
3870:An Introduction to Measure Theory
3323:As a remark, all of the measures
1284:{\displaystyle k\in \mathbb {N} }
688:{\displaystyle m\in \mathbb {N} }
157:{\displaystyle k\in \mathbb {N} }
129:{\displaystyle n\in \mathbb {N} }
2377:{\displaystyle \mathbb {R} ^{n}}
2348:{\displaystyle \mathbb {R} ^{n}}
1704:{\displaystyle t_{1}\dots t_{k}}
1433:{\displaystyle t_{1}\dots t_{k}}
52:finite-dimensional distributions
3875:Graduate Studies in Mathematics
3891:
3857:
3830:
3682:
3666:
3628:
3614:
3272:
3253:
3091:
3072:
3033:
3014:
2761:
2753:
2720:
2451:
2421:
2277:
2237:
2226:
2207:
2125:
2095:
2073:
2043:
2007:
1977:
1941:
1911:
1875:
1837:
1778:
1762:
1605:
1589:
1539:
1536:
1524:
1472:
1456:
1026:
991:
967:
527:
521:
499:
493:
473:
467:
451:
445:
373:{\displaystyle \{1,\dots ,k\}}
294:
278:
40:Kolmogorov consistency theorem
1:
3823:
3349:{\displaystyle \mu _{F},\mu }
2316:Kolmogorov continuity theorem
1801:Explanation of the conditions
70:Andrey Nikolaevich Kolmogorov
3176:{\displaystyle \pi _{F}^{G}}
2952:{\displaystyle \{\mu _{F}\}}
36:Kolmogorov existence theorem
32:Kolmogorov extension theorem
7:
3789:Brownian motion, i.e., the
3226:{\displaystyle \Omega _{T}}
2919:{\displaystyle \Omega _{F}}
2857:{\displaystyle \Omega _{F}}
2557:{\displaystyle \Omega _{t}}
2322:General form of the theorem
2294:Implications of the theorem
625:2. for all measurable sets
10:
3947:
3807:
3313:{\displaystyle F\subset T}
2803:{\displaystyle F\subset T}
2583:{\displaystyle J\subset T}
1888:can be computed either as
1797:, which is not very rich.
1256:{\displaystyle t_{i}\in T}
44:Daniell-Kolmogorov theorem
18:
3701:with probability measure
2892:{\displaystyle \tau _{t}}
2564:. For each finite subset
2526:{\displaystyle \tau _{t}}
1004:and a stochastic process
3837:Ćksendal, Bernt (2003).
3294:for every finite subset
3144:{\displaystyle \mu _{G}}
2830:{\displaystyle \mu _{F}}
76:Statement of the theorem
2784:For each finite subset
2165:, and any control sets
1825:. Then the probability
1751:is only defined on the
21:HahnāKolmogorov theorem
3768:
3721:
3720:{\displaystyle P=\mu }
3695:
3647:
3601:
3543:
3445:
3444:{\displaystyle t\in T}
3419:
3374:
3350:
3314:
3288:
3227:
3200:
3177:
3145:
3114:
3056:
2985:
2953:
2920:
2893:
2858:
2831:
2804:
2775:
2737:
2679:
2643:
2584:
2558:
2527:
2500:
2499:{\displaystyle t\in T}
2474:
2405:
2386:inner regular measures
2378:
2349:
2284:
2186:
2159:
2132:
2014:
1948:
1882:
1819:
1791:
1745:
1725:
1705:
1665:
1618:
1576:
1556:
1511:the canonical process
1505:
1485:
1434:
1394:
1347:
1327:
1285:
1257:
1221:
1045:
998:
948:
689:
661:
616:
416:
374:
336:
310:
261:
214:
158:
130:
94:
3769:
3722:
3696:
3648:
3602:
3544:
3446:
3420:
3375:
3358:product sigma algebra
3351:
3315:
3289:
3228:
3201:
3178:
3146:
3115:
3057:
2986:
2954:
2921:
2894:
2859:
2832:
2805:
2776:
2738:
2680:
2644:
2585:
2559:
2528:
2501:
2475:
2406:
2379:
2350:
2285:
2187:
2185:{\displaystyle F_{i}}
2160:
2158:{\displaystyle t_{i}}
2133:
2015:
1949:
1883:
1820:
1792:
1746:
1726:
1706:
1666:
1619:
1577:
1557:
1506:
1486:
1435:
1395:
1348:
1328:
1286:
1258:
1222:
1046:
999:
949:
690:
662:
617:
417:
375:
337:
311:
262:
215:
159:
131:
95:
3731:
3705:
3657:
3611:
3553:
3455:
3429:
3387:
3373:{\displaystyle \mu }
3364:
3327:
3298:
3237:
3210:
3199:{\displaystyle \mu }
3190:
3155:
3128:
3069:
2998:
2963:
2930:
2903:
2876:
2868:with respect to the
2841:
2814:
2788:
2747:
2689:
2657:
2597:
2568:
2541:
2510:
2484:
2415:
2395:
2359:
2330:
2199:
2169:
2142:
2024:
1958:
1892:
1829:
1809:
1759:
1744:{\displaystyle \nu }
1735:
1715:
1675:
1628:
1586:
1575:{\displaystyle \nu }
1566:
1515:
1495:
1447:
1404:
1357:
1337:
1295:
1291:and measurable sets
1267:
1234:
1058:
1008:
964:
957:Then there exists a
701:
671:
629:
429:
384:
380:and measurable sets
346:
335:{\displaystyle \pi }
326:
275:
224:
172:
140:
112:
84:
3910:Aldrich, J. (2007)
3356:are defined on the
3270:
3172:
3122:pushforward measure
3089:
3031:
2706:
269:probability measure
3818:Percy John Daniell
3778:are automatically
3764:
3717:
3691:
3643:
3597:
3539:
3441:
3415:
3370:
3346:
3310:
3284:
3256:
3223:
3196:
3173:
3158:
3141:
3110:
3075:
3052:
3017:
2981:
2949:
2916:
2889:
2854:
2827:
2800:
2771:
2733:
2692:
2675:
2639:
2628:
2580:
2554:
2535:Hausdorff topology
2523:
2496:
2470:
2401:
2374:
2345:
2280:
2182:
2155:
2128:
2010:
1944:
1878:
1815:
1787:
1741:
1721:
1701:
1661:
1614:
1572:
1552:
1501:
1481:
1430:
1390:
1343:
1323:
1281:
1253:
1217:
1041:
994:
944:
935:
928:
685:
657:
612:
412:
370:
332:
306:
257:
210:
168:of distinct times
154:
126:
90:
60:Percy John Daniell
56:stochastic process
3884:978-0-8218-6919-2
2613:
2404:{\displaystyle T}
1818:{\displaystyle X}
1753:product Ļ-algebra
1724:{\displaystyle T}
1504:{\displaystyle X}
1346:{\displaystyle X}
959:probability space
889:
887:
93:{\displaystyle T}
3938:
3898:
3895:
3889:
3888:
3861:
3855:
3854:
3834:
3773:
3771:
3770:
3765:
3763:
3762:
3761:
3760:
3748:
3747:
3726:
3724:
3723:
3718:
3700:
3698:
3697:
3692:
3690:
3689:
3680:
3679:
3674:
3652:
3650:
3649:
3644:
3642:
3641:
3626:
3625:
3606:
3604:
3603:
3598:
3590:
3589:
3565:
3564:
3548:
3546:
3545:
3540:
3538:
3537:
3536:
3535:
3523:
3522:
3505:
3504:
3500:
3499:
3475:
3474:
3450:
3448:
3447:
3442:
3424:
3422:
3421:
3416:
3414:
3413:
3408:
3399:
3398:
3379:
3377:
3376:
3371:
3355:
3353:
3352:
3347:
3339:
3338:
3319:
3317:
3316:
3311:
3293:
3291:
3290:
3285:
3280:
3279:
3269:
3264:
3249:
3248:
3232:
3230:
3229:
3224:
3222:
3221:
3205:
3203:
3202:
3197:
3182:
3180:
3179:
3174:
3171:
3166:
3150:
3148:
3147:
3142:
3140:
3139:
3119:
3117:
3116:
3111:
3109:
3108:
3099:
3098:
3088:
3083:
3061:
3059:
3058:
3053:
3051:
3050:
3041:
3040:
3030:
3025:
3010:
3009:
2990:
2988:
2987:
2982:
2958:
2956:
2955:
2950:
2945:
2944:
2925:
2923:
2922:
2917:
2915:
2914:
2898:
2896:
2895:
2890:
2888:
2887:
2872:(induced by the
2870:product topology
2863:
2861:
2860:
2855:
2853:
2852:
2836:
2834:
2833:
2828:
2826:
2825:
2809:
2807:
2806:
2801:
2780:
2778:
2777:
2772:
2770:
2769:
2764:
2742:
2740:
2739:
2734:
2732:
2731:
2719:
2718:
2705:
2700:
2684:
2682:
2681:
2676:
2648:
2646:
2645:
2640:
2638:
2637:
2627:
2609:
2608:
2589:
2587:
2586:
2581:
2563:
2561:
2560:
2555:
2553:
2552:
2532:
2530:
2529:
2524:
2522:
2521:
2505:
2503:
2502:
2497:
2479:
2477:
2476:
2471:
2469:
2468:
2450:
2449:
2444:
2443:
2433:
2432:
2411:be any set. Let
2410:
2408:
2407:
2402:
2383:
2381:
2380:
2375:
2373:
2372:
2367:
2354:
2352:
2351:
2346:
2344:
2343:
2338:
2289:
2287:
2286:
2281:
2276:
2268:
2267:
2249:
2248:
2236:
2219:
2218:
2206:
2191:
2189:
2188:
2183:
2181:
2180:
2164:
2162:
2161:
2156:
2154:
2153:
2137:
2135:
2134:
2129:
2124:
2123:
2118:
2109:
2108:
2103:
2094:
2093:
2072:
2071:
2066:
2057:
2056:
2051:
2042:
2041:
2019:
2017:
2016:
2011:
2006:
2005:
2000:
1991:
1990:
1985:
1976:
1975:
1953:
1951:
1950:
1945:
1940:
1939:
1934:
1925:
1924:
1919:
1910:
1909:
1887:
1885:
1884:
1879:
1868:
1867:
1849:
1848:
1836:
1824:
1822:
1821:
1816:
1796:
1794:
1793:
1788:
1786:
1785:
1776:
1775:
1770:
1750:
1748:
1747:
1742:
1730:
1728:
1727:
1722:
1710:
1708:
1707:
1702:
1700:
1699:
1687:
1686:
1670:
1668:
1667:
1662:
1660:
1659:
1658:
1657:
1645:
1644:
1623:
1621:
1620:
1615:
1613:
1612:
1603:
1602:
1597:
1581:
1579:
1578:
1573:
1561:
1559:
1558:
1553:
1551:
1550:
1510:
1508:
1507:
1502:
1491:and to take for
1490:
1488:
1487:
1482:
1480:
1479:
1470:
1469:
1464:
1439:
1437:
1436:
1431:
1429:
1428:
1416:
1415:
1399:
1397:
1396:
1391:
1389:
1388:
1387:
1386:
1374:
1373:
1352:
1350:
1349:
1344:
1332:
1330:
1329:
1324:
1322:
1321:
1316:
1307:
1306:
1290:
1288:
1287:
1282:
1280:
1262:
1260:
1259:
1254:
1246:
1245:
1226:
1224:
1223:
1218:
1216:
1212:
1211:
1210:
1198:
1197:
1196:
1195:
1172:
1171:
1159:
1158:
1157:
1156:
1137:
1129:
1125:
1124:
1123:
1105:
1104:
1090:
1089:
1088:
1087:
1075:
1074:
1050:
1048:
1047:
1042:
1040:
1039:
1034:
1003:
1001:
1000:
995:
990:
982:
981:
953:
951:
950:
945:
940:
936:
934:
929:
924:
923:
922:
917:
902:
901:
896:
883:
882:
864:
863:
849:
848:
847:
846:
822:
821:
803:
802:
790:
789:
772:
768:
767:
766:
748:
747:
733:
732:
731:
730:
718:
717:
694:
692:
691:
686:
684:
666:
664:
663:
658:
656:
655:
650:
641:
640:
621:
619:
618:
613:
608:
604:
603:
602:
584:
583:
569:
568:
567:
566:
554:
553:
536:
532:
531:
530:
503:
502:
479:
478:
477:
476:
455:
454:
421:
419:
418:
413:
411:
410:
405:
396:
395:
379:
377:
376:
371:
341:
339:
338:
333:
315:
313:
312:
307:
302:
301:
292:
291:
286:
266:
264:
263:
258:
256:
255:
254:
253:
241:
240:
219:
217:
216:
211:
203:
202:
184:
183:
163:
161:
160:
155:
153:
135:
133:
132:
127:
125:
104:(thought of as "
99:
97:
96:
91:
3946:
3945:
3941:
3940:
3939:
3937:
3936:
3935:
3921:
3920:
3907:
3902:
3901:
3896:
3892:
3885:
3862:
3858:
3851:
3835:
3831:
3826:
3810:
3756:
3752:
3743:
3739:
3738:
3734:
3732:
3729:
3728:
3706:
3703:
3702:
3685:
3681:
3675:
3670:
3669:
3658:
3655:
3654:
3631:
3627:
3621:
3617:
3612:
3609:
3608:
3585:
3581:
3560:
3556:
3554:
3551:
3550:
3531:
3527:
3518:
3514:
3513:
3509:
3495:
3491:
3470:
3466:
3462:
3458:
3456:
3453:
3452:
3430:
3427:
3426:
3409:
3404:
3403:
3394:
3390:
3388:
3385:
3384:
3365:
3362:
3361:
3334:
3330:
3328:
3325:
3324:
3299:
3296:
3295:
3275:
3271:
3265:
3260:
3244:
3240:
3238:
3235:
3234:
3217:
3213:
3211:
3208:
3207:
3191:
3188:
3187:
3167:
3162:
3156:
3153:
3152:
3135:
3131:
3129:
3126:
3125:
3104:
3100:
3094:
3090:
3084:
3079:
3070:
3067:
3066:
3046:
3042:
3036:
3032:
3026:
3021:
3005:
3001:
2999:
2996:
2995:
2991:, we have that
2964:
2961:
2960:
2940:
2936:
2931:
2928:
2927:
2910:
2906:
2904:
2901:
2900:
2883:
2879:
2877:
2874:
2873:
2848:
2844:
2842:
2839:
2838:
2821:
2817:
2815:
2812:
2811:
2789:
2786:
2785:
2765:
2760:
2759:
2748:
2745:
2744:
2727:
2723:
2714:
2710:
2701:
2696:
2690:
2687:
2686:
2658:
2655:
2654:
2633:
2629:
2617:
2604:
2600:
2598:
2595:
2594:
2569:
2566:
2565:
2548:
2544:
2542:
2539:
2538:
2517:
2513:
2511:
2508:
2507:
2485:
2482:
2481:
2458:
2454:
2445:
2439:
2438:
2437:
2428:
2424:
2416:
2413:
2412:
2396:
2393:
2392:
2368:
2363:
2362:
2360:
2357:
2356:
2339:
2334:
2333:
2331:
2328:
2327:
2324:
2312:Brownian motion
2308:Brownian motion
2296:
2272:
2263:
2259:
2244:
2240:
2232:
2214:
2210:
2202:
2200:
2197:
2196:
2176:
2172:
2170:
2167:
2166:
2149:
2145:
2143:
2140:
2139:
2119:
2114:
2113:
2104:
2099:
2098:
2083:
2079:
2067:
2062:
2061:
2052:
2047:
2046:
2031:
2027:
2025:
2022:
2021:
2001:
1996:
1995:
1986:
1981:
1980:
1965:
1961:
1959:
1956:
1955:
1935:
1930:
1929:
1920:
1915:
1914:
1899:
1895:
1893:
1890:
1889:
1863:
1859:
1844:
1840:
1832:
1830:
1827:
1826:
1810:
1807:
1806:
1803:
1781:
1777:
1771:
1766:
1765:
1760:
1757:
1756:
1736:
1733:
1732:
1716:
1713:
1712:
1695:
1691:
1682:
1678:
1676:
1673:
1672:
1653:
1649:
1640:
1636:
1635:
1631:
1629:
1626:
1625:
1624:with marginals
1608:
1604:
1598:
1593:
1592:
1587:
1584:
1583:
1567:
1564:
1563:
1546:
1542:
1516:
1513:
1512:
1496:
1493:
1492:
1475:
1471:
1465:
1460:
1459:
1448:
1445:
1444:
1424:
1420:
1411:
1407:
1405:
1402:
1401:
1382:
1378:
1369:
1365:
1364:
1360:
1358:
1355:
1354:
1338:
1335:
1334:
1317:
1312:
1311:
1302:
1298:
1296:
1293:
1292:
1276:
1268:
1265:
1264:
1241:
1237:
1235:
1232:
1231:
1206:
1202:
1191:
1187:
1186:
1182:
1167:
1163:
1152:
1148:
1147:
1143:
1142:
1138:
1133:
1119:
1115:
1100:
1096:
1095:
1091:
1083:
1079:
1070:
1066:
1065:
1061:
1059:
1056:
1055:
1035:
1030:
1029:
1009:
1006:
1005:
986:
977:
976:
965:
962:
961:
930:
918:
913:
912:
897:
892:
891:
890:
888:
878:
874:
859:
855:
854:
850:
836:
832:
811:
807:
798:
794:
785:
781:
780:
776:
762:
758:
743:
739:
738:
734:
726:
722:
713:
709:
708:
704:
702:
699:
698:
680:
672:
669:
668:
651:
646:
645:
636:
632:
630:
627:
626:
598:
594:
579:
575:
574:
570:
562:
558:
549:
545:
544:
540:
517:
513:
489:
485:
484:
480:
463:
459:
441:
437:
436:
432:
430:
427:
426:
406:
401:
400:
391:
387:
385:
382:
381:
347:
344:
343:
327:
324:
323:
297:
293:
287:
282:
281:
276:
273:
272:
249:
245:
236:
232:
231:
227:
225:
222:
221:
198:
194:
179:
175:
173:
170:
169:
149:
141:
138:
137:
121:
113:
110:
109:
85:
82:
81:
78:
34:(also known as
24:
17:
12:
11:
5:
3944:
3934:
3933:
3919:
3918:
3917:December 2007.
3906:
3905:External links
3903:
3900:
3899:
3890:
3883:
3856:
3849:
3828:
3827:
3825:
3822:
3816:mathematician
3809:
3806:
3805:
3804:
3801:
3794:
3791:Wiener process
3759:
3755:
3751:
3746:
3742:
3737:
3716:
3713:
3710:
3688:
3684:
3678:
3673:
3668:
3665:
3662:
3640:
3637:
3634:
3630:
3624:
3620:
3616:
3596:
3593:
3588:
3584:
3580:
3577:
3574:
3571:
3568:
3563:
3559:
3534:
3530:
3526:
3521:
3517:
3512:
3508:
3503:
3498:
3494:
3490:
3487:
3484:
3481:
3478:
3473:
3469:
3465:
3461:
3440:
3437:
3434:
3412:
3407:
3402:
3397:
3393:
3369:
3345:
3342:
3337:
3333:
3309:
3306:
3303:
3283:
3278:
3274:
3268:
3263:
3259:
3255:
3252:
3247:
3243:
3220:
3216:
3195:
3170:
3165:
3161:
3138:
3134:
3107:
3103:
3097:
3093:
3087:
3082:
3078:
3074:
3063:
3062:
3049:
3045:
3039:
3035:
3029:
3024:
3020:
3016:
3013:
3008:
3004:
2980:
2977:
2974:
2971:
2968:
2948:
2943:
2939:
2935:
2913:
2909:
2886:
2882:
2851:
2847:
2824:
2820:
2799:
2796:
2793:
2768:
2763:
2758:
2755:
2752:
2730:
2726:
2722:
2717:
2713:
2709:
2704:
2699:
2695:
2674:
2671:
2668:
2665:
2662:
2651:
2650:
2636:
2632:
2626:
2623:
2620:
2616:
2612:
2607:
2603:
2579:
2576:
2573:
2551:
2547:
2520:
2516:
2495:
2492:
2489:
2467:
2464:
2461:
2457:
2453:
2448:
2442:
2436:
2431:
2427:
2423:
2420:
2400:
2371:
2366:
2342:
2337:
2323:
2320:
2295:
2292:
2279:
2275:
2271:
2266:
2262:
2258:
2255:
2252:
2247:
2243:
2239:
2235:
2231:
2228:
2225:
2222:
2217:
2213:
2209:
2205:
2179:
2175:
2152:
2148:
2127:
2122:
2117:
2112:
2107:
2102:
2097:
2092:
2089:
2086:
2082:
2078:
2075:
2070:
2065:
2060:
2055:
2050:
2045:
2040:
2037:
2034:
2030:
2009:
2004:
1999:
1994:
1989:
1984:
1979:
1974:
1971:
1968:
1964:
1943:
1938:
1933:
1928:
1923:
1918:
1913:
1908:
1905:
1902:
1898:
1877:
1874:
1871:
1866:
1862:
1858:
1855:
1852:
1847:
1843:
1839:
1835:
1814:
1802:
1799:
1784:
1780:
1774:
1769:
1764:
1740:
1720:
1698:
1694:
1690:
1685:
1681:
1656:
1652:
1648:
1643:
1639:
1634:
1611:
1607:
1601:
1596:
1591:
1571:
1549:
1545:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1500:
1478:
1474:
1468:
1463:
1458:
1455:
1452:
1427:
1423:
1419:
1414:
1410:
1385:
1381:
1377:
1372:
1368:
1363:
1342:
1320:
1315:
1310:
1305:
1301:
1279:
1275:
1272:
1252:
1249:
1244:
1240:
1228:
1227:
1215:
1209:
1205:
1201:
1194:
1190:
1185:
1181:
1178:
1175:
1170:
1166:
1162:
1155:
1151:
1146:
1141:
1136:
1132:
1128:
1122:
1118:
1114:
1111:
1108:
1103:
1099:
1094:
1086:
1082:
1078:
1073:
1069:
1064:
1038:
1033:
1028:
1025:
1022:
1019:
1016:
1013:
993:
989:
985:
980:
975:
972:
969:
955:
954:
943:
939:
933:
927:
921:
916:
911:
908:
905:
900:
895:
886:
881:
877:
873:
870:
867:
862:
858:
853:
845:
842:
839:
835:
831:
828:
825:
820:
817:
814:
810:
806:
801:
797:
793:
788:
784:
779:
775:
771:
765:
761:
757:
754:
751:
746:
742:
737:
729:
725:
721:
716:
712:
707:
683:
679:
676:
654:
649:
644:
639:
635:
623:
622:
611:
607:
601:
597:
593:
590:
587:
582:
578:
573:
565:
561:
557:
552:
548:
543:
539:
535:
529:
526:
523:
520:
516:
512:
509:
506:
501:
498:
495:
492:
488:
483:
475:
472:
469:
466:
462:
458:
453:
450:
447:
444:
440:
435:
409:
404:
399:
394:
390:
369:
366:
363:
360:
357:
354:
351:
331:
305:
300:
296:
290:
285:
280:
252:
248:
244:
239:
235:
230:
209:
206:
201:
197:
193:
190:
187:
182:
178:
152:
148:
145:
124:
120:
117:
89:
77:
74:
54:will define a
15:
9:
6:
4:
3:
2:
3943:
3932:
3929:
3928:
3926:
3916:
3913:
3909:
3908:
3894:
3886:
3880:
3876:
3872:
3871:
3866:
3860:
3852:
3850:3-540-04758-1
3846:
3842:
3841:
3833:
3829:
3821:
3819:
3815:
3802:
3799:
3795:
3792:
3788:
3787:
3786:
3783:
3781:
3777:
3776:Polish spaces
3757:
3753:
3749:
3744:
3740:
3735:
3714:
3711:
3708:
3686:
3676:
3663:
3653:, defined on
3638:
3635:
3632:
3622:
3618:
3594:
3591:
3586:
3582:
3578:
3575:
3572:
3569:
3566:
3561:
3557:
3532:
3528:
3524:
3519:
3515:
3510:
3506:
3496:
3492:
3488:
3485:
3482:
3479:
3476:
3471:
3467:
3459:
3438:
3435:
3432:
3410:
3400:
3395:
3381:
3367:
3359:
3343:
3340:
3335:
3331:
3321:
3307:
3304:
3301:
3281:
3276:
3266:
3261:
3257:
3250:
3245:
3241:
3218:
3193:
3184:
3168:
3163:
3159:
3136:
3132:
3123:
3105:
3101:
3095:
3085:
3080:
3076:
3047:
3043:
3037:
3027:
3022:
3018:
3011:
3006:
3002:
2994:
2993:
2992:
2978:
2975:
2972:
2969:
2966:
2941:
2937:
2911:
2884:
2880:
2871:
2867:
2866:inner regular
2849:
2822:
2818:
2797:
2794:
2791:
2782:
2766:
2756:
2750:
2728:
2715:
2707:
2702:
2697:
2693:
2672:
2669:
2666:
2663:
2660:
2634:
2624:
2621:
2618:
2614:
2610:
2605:
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2577:
2574:
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2536:
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2490:
2487:
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2462:
2459:
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2398:
2389:
2387:
2369:
2340:
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2317:
2313:
2309:
2304:
2300:
2291:
2269:
2264:
2260:
2256:
2253:
2250:
2245:
2241:
2229:
2223:
2220:
2215:
2211:
2193:
2177:
2173:
2150:
2146:
2120:
2110:
2105:
2090:
2087:
2084:
2080:
2076:
2068:
2058:
2053:
2038:
2035:
2032:
2028:
2002:
1992:
1987:
1972:
1969:
1966:
1962:
1936:
1926:
1921:
1906:
1903:
1900:
1896:
1872:
1869:
1864:
1860:
1856:
1853:
1850:
1845:
1841:
1812:
1798:
1782:
1772:
1754:
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1718:
1696:
1692:
1688:
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1646:
1641:
1637:
1632:
1609:
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1569:
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1543:
1533:
1530:
1527:
1521:
1518:
1498:
1476:
1466:
1453:
1441:
1425:
1421:
1417:
1412:
1408:
1383:
1379:
1375:
1370:
1366:
1361:
1340:
1318:
1308:
1303:
1299:
1273:
1270:
1250:
1247:
1242:
1238:
1213:
1207:
1203:
1199:
1192:
1188:
1183:
1179:
1176:
1173:
1168:
1164:
1160:
1153:
1149:
1144:
1139:
1130:
1126:
1120:
1116:
1112:
1109:
1106:
1101:
1097:
1092:
1084:
1080:
1076:
1071:
1067:
1062:
1054:
1053:
1052:
1036:
1020:
1017:
1014:
1011:
983:
973:
960:
941:
937:
931:
925:
919:
909:
906:
903:
898:
884:
879:
875:
871:
868:
865:
860:
856:
851:
843:
840:
837:
833:
829:
826:
823:
818:
815:
812:
808:
804:
799:
795:
791:
786:
782:
777:
773:
769:
763:
759:
755:
752:
749:
744:
740:
735:
727:
723:
719:
714:
710:
705:
697:
696:
695:
677:
674:
652:
642:
637:
633:
609:
605:
599:
595:
591:
588:
585:
580:
576:
571:
563:
559:
555:
550:
546:
541:
537:
533:
524:
518:
514:
510:
507:
504:
496:
490:
486:
481:
470:
464:
460:
456:
448:
442:
438:
433:
425:
424:
423:
407:
397:
392:
388:
364:
361:
358:
355:
352:
329:
322:
317:
303:
298:
288:
270:
250:
246:
242:
237:
233:
228:
207:
204:
199:
195:
191:
188:
185:
180:
176:
167:
146:
143:
118:
115:
107:
103:
87:
73:
71:
68:
67:mathematician
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
22:
3893:
3869:
3859:
3839:
3832:
3811:
3798:Markov chain
3784:
3382:
3322:
3185:
3120:denotes the
3064:
2783:
2653:For subsets
2652:
2390:
2325:
2305:
2301:
2297:
2194:
1804:
1442:
1229:
956:
624:
321:permutations
318:
108:"), and let
100:denote some
79:
43:
39:
35:
31:
25:
319:1. for all
164:and finite
136:. For each
28:mathematics
3824:References
3233:such that
1051:such that
3750:…
3736:ν
3715:μ
3661:Ω
3636:∈
3619:π
3592:∈
3525:…
3511:ν
3460:μ
3436:∈
3392:Ω
3368:μ
3344:μ
3332:μ
3305:⊂
3282:μ
3277:∗
3258:π
3242:μ
3215:Ω
3194:μ
3160:π
3133:μ
3102:μ
3096:∗
3077:π
3044:μ
3038:∗
3019:π
3003:μ
2976:⊂
2970:⊂
2938:μ
2908:Ω
2881:τ
2864:which is
2846:Ω
2819:μ
2795:⊂
2757:ω
2754:↦
2751:ω
2725:Ω
2721:→
2712:Ω
2694:π
2670:⊂
2664:⊂
2631:Ω
2622:∈
2615:∏
2602:Ω
2590:, define
2575:⊂
2546:Ω
2515:τ
2491:∈
2463:∈
2426:Ω
2270:∈
2111:×
2106:−
2081:ν
2069:−
2059:×
2029:ν
1993:×
1988:−
1963:ν
1937:−
1927:×
1897:ν
1739:ν
1689:…
1647:…
1633:ν
1570:ν
1540:↦
1522::
1451:Ω
1418:…
1376:…
1362:ν
1309:⊆
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1200:∈
1177:…
1161:∈
1113:×
1110:⋯
1107:×
1077:…
1063:ν
1027:→
1024:Ω
1021:×
971:Ω
926:⏟
910:×
907:⋯
904:×
885:×
872:×
869:⋯
866:×
827:…
792:…
778:ν
756:×
753:⋯
750:×
720:…
706:ν
678:∈
643:⊆
592:×
589:⋯
586:×
556:…
542:ν
519:π
511:×
508:⋯
505:×
491:π
465:π
457:…
443:π
434:ν
398:⊆
359:…
330:π
243:…
229:ν
205:∈
189:…
147:∈
119:∈
3925:Category
3867:(2011).
3425:for all
1230:for all
166:sequence
102:interval
62:and the
3865:Tao, T.
3814:British
3808:History
1954:or as
1333:, i.e.
64:Russian
48:theorem
46:) is a
42:or the
3881:
3847:
3451:, and
3065:where
2685:, let
2506:, let
220:, let
38:, the
30:, the
3780:Radon
2899:) on
2533:be a
267:be a
3879:ISBN
3845:ISBN
3549:for
2391:Let
2251:>
2221:>
1870:<
1851:>
1353:has
106:time
80:Let
3206:on
3124:of
2837:on
2537:on
1755:of
1582:on
342:of
271:on
26:In
3927::
3873:.
3796:a
3782:.
3320:.
3183:.
2781:.
2611::=
2192:.
1440:.
1263:,
422:,
72:.
3887:.
3853:.
3793:,
3758:k
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3712:=
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3683:)
3677:n
3672:R
3667:(
3664:=
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3576:.
3573:.
3570:.
3567:,
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3486:.
3483:.
3480:.
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3336:F
3308:T
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3267:T
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3254:(
3251:=
3246:F
3219:T
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3164:F
3137:G
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3086:G
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3073:(
3048:G
3034:)
3028:G
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3015:(
3012:=
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2729:I
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2649:.
2635:t
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2447:t
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2430:t
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2419:{
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2257:,
2254:0
2246:1
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2238:(
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2224:0
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2208:(
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2121:+
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2101:R
2096:(
2091:1
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2064:R
2054:+
2049:R
2044:(
2039:2
2036:,
2033:1
2008:)
2003:+
1998:R
1983:R
1978:(
1973:1
1970:,
1967:2
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1932:R
1922:+
1917:R
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1901:1
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1873:0
1865:2
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1600:n
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938:)
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736:(
728:k
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682:N
675:m
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23:.
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