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is a tail event. Thus by
Kolmogorov 0-1 law, it has either probability 0 or 1 to happen. Note that independence is required for the tail event condition to hold. Without independence we can consider a sequence that's either
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60:
Tail events are defined in terms of countably infinite families of σ-algebras. For illustrative purposes, we present here the special case in which each sigma algebra is generated by a random variable
966:
1398:
Curriculum Vitae and
Biography. Kolmogorov School. Ph.D. students and descendants of A. N. Kolmogorov. A. N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A. N. Kolmogorov.
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501:-th coin toss in a modeled, infinite sequence of coin tosses, this means that a sequence of 100 consecutive heads occurring infinitely many times is a tail event in this model.
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In many situations, it can be easy to apply
Kolmogorov's zero–one law to show that some event has probability 0 or 1, but surprisingly hard to determine
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295:, but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) For example, the event that the sequence of the
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Tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the
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898:. A tail event is then by definition an event which is measurable with respect to the σ-algebra generated by all
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A more general statement of
Kolmogorov's zero–one law holds for sequences of independent σ-algebras. Let (Ω,
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The statement of the law in terms of random variables is obtained from the latter by taking each
1118:{\displaystyle \left\{\lim _{n\rightarrow \infty }\sum _{k=1}^{n}X_{k}{\text{ exists }}\right\}}
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are, for example, all
Bernoulli-distributed, then the event that there are infinitely many
325:
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143:
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8:
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775:{\displaystyle {\mathcal {T}}((F_{n})_{n\in \mathbb {N} })=\bigcap _{n=1}^{\infty }G_{n}}
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251:
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27:
1375:
1348:
1324:
555:
916:. That is, a tail event is precisely an element of the terminal σ-algebra
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converges, and the event that its sum converges are both tail events. If the
50:
45:
645:{\displaystyle G_{n}=\sigma {\bigg (}\bigcup _{k=n}^{\infty }F_{k}{\bigg )}}
19:"Tail event" redirects here. For "tail events" meaning "rare events", see
54:
977:
1302:
1297:
1370:. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. p.
20:
854:{\displaystyle E\in {\mathcal {T}}((F_{n})_{n\in \mathbb {N} })}
1046:
be a sequence of independent random variables, then the event
961:{\displaystyle \textstyle {\bigcap _{n=1}^{\infty }G_{n}}}
889:
to be the σ-algebra generated by the random variable
16:
Special case in probability theory; introduces tail events
999:. The presence of an infinite cluster in the context of
204:
of each finite subset of these random variables. (Note:
1241:
each. In this case the sum converges with probability
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be the sigma-algebra generated jointly by all of the
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907:, but which is independent of any finite number of
794:are stochastically independent, then for any event
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1339:Brzezniak, Zdzislaw; Zastawniak, Thomasz (2000).
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53:happen or almost surely not happen; that is, the
1403:
1059:
538:of these two extreme values is the correct one.
447:{\displaystyle X_{k}=X_{k+1}=\dots =X_{k+100}=1}
785:Kolmogorov's zero–one law asserts that, if the
567:be a sequence of σ-algebras contained in
1027:
1013:
268:is uniquely determined by the values of the
1396:The Legacy of Andrei Nikolaevich Kolmogorov
1366:A first look at rigorous probability theory
57:of such an event occurring is zero or one.
655:be the smallest σ-algebra containing
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842:
729:
363:
102:
1314:
1404:
995:are Kolmogorov automorphisms but not
1317:Probability theory: An analytic view
987:that obeys the 0-1 law is called a
193:{\displaystyle F\in {\mathcal {F}}}
38:, specifies that a certain type of
13:
1069:
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809:
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14:
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981:measure-preserving transformation
370:{\displaystyle k\in \mathbb {N} }
109:{\displaystyle k\in \mathbb {N} }
1362:Rosenthal, Jeffrey S. (2006).
1261:{\displaystyle {\frac {1}{2}}}
1234:{\displaystyle {\frac {1}{2}}}
1207:{\displaystyle (1,1,1,\dots )}
1201:
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1163:{\displaystyle (0,0,0,\dots )}
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241:{\displaystyle {\mathcal {F}}}
133:{\displaystyle {\mathcal {F}}}
1:
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1039:{\displaystyle \{X_{n}\}_{n}}
202:probabilistically independent
36:Andrey Nikolaevich Kolmogorov
7:
1271:
971:
248:implies that membership in
10:
1433:
1341:Basic Stochastic Processes
1321:Cambridge University Press
1283:Hewitt–Savage zero–one law
985:standard probability space
481:models the outcome of the
44:tail event of independent
18:
454:is a tail event. If each
32:Kolmogorov's zero–one law
1315:Stroock, Daniel (1999).
1003:also obeys the 0-1 law.
993:Bernoulli automorphisms
989:Kolmogorov automorphism
676:terminal σ-algebra
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524:{\displaystyle X_{k}}
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342:{\displaystyle X_{k}}
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200:is an event which is
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160:{\displaystyle X_{k}}
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111:
83:
81:{\displaystyle X_{k}}
1412:Probability theorems
1319:(revised ed.).
1278:Borel–Cantelli lemma
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34:, named in honor of
1288:Lévy's zero–one law
1293:Tail sigma-algebra
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1001:percolation theory
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28:probability theory
1381:978-981-270-371-2
1330:978-0-521-66349-6
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1214:with probability
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861:, one has either
556:probability space
494:{\displaystyle k}
261:{\displaystyle F}
217:{\displaystyle F}
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1417:Covering lemmas
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46:σ-algebras
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1390:External links
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1354:3-540-76175-6
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224:belonging to
211:
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95:
73:
69:
58:
56:
52:
51:almost surely
48:
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41:
37:
33:
29:
22:
1365:
1340:
1316:
1005:
996:
975:
912:
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862:
790:
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675:
669:
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660:
656:
654:
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559:
551:
547:
545:
535:
533:
531:is removed.
503:
168:
59:
43:
31:
25:
674:, .... The
542:Formulation
55:probability
42:, namely a
1406:Categories
1309:References
997:vice versa
978:invertible
377:such that
169:tail event
167:. Then, a
1303:Tail risk
1298:Long tail
1199:…
1155:…
1076:∑
1070:∞
1067:→
942:∞
927:⋂
869:) = 0 or
839:∈
805:∈
758:∞
743:⋂
726:∈
621:∞
606:⋃
595:σ
417:⋯
360:∈
181:∈
99:∈
1345:Springer
1272:See also
972:Examples
558:and let
21:fat tail
678:of the
554:) be a
1378:
1351:
1327:
991:. All
571:. Let
116:. Let
983:on a
877:)=1.
536:which
40:event
1376:ISBN
1349:ISBN
1325:ISBN
1006:Let
88:for
1170:or
1060:lim
976:An
434:100
26:In
1408::
1374:.
1372:37
1347:.
1343:.
1323:.
1268:.
968:.
782:.
672:+1
664:,
30:,
1384:.
1357:.
1335:.
1333:.
1254:2
1251:1
1227:2
1224:1
1202:)
1196:,
1193:1
1190:,
1187:1
1184:,
1181:1
1178:(
1158:)
1152:,
1149:0
1146:,
1143:0
1140:,
1137:0
1134:(
1112:}
1101:k
1097:X
1091:n
1086:1
1083:=
1080:k
1064:n
1055:{
1032:n
1028:}
1022:n
1018:X
1014:{
952:n
948:G
937:1
934:=
931:n
913:n
909:X
904:n
900:X
895:n
891:X
886:n
882:F
875:E
873:(
871:P
867:E
865:(
863:P
849:)
843:N
836:n
832:)
826:n
822:F
818:(
815:(
810:T
802:E
791:n
787:F
768:n
764:G
753:1
750:=
747:n
739:=
736:)
730:N
723:n
719:)
713:n
709:F
705:(
702:(
697:T
684:n
680:F
670:n
666:F
661:n
657:F
638:)
631:k
627:F
616:n
613:=
610:k
600:(
592:=
587:n
583:G
569:F
564:n
560:F
552:P
550:,
548:F
517:k
513:X
489:k
467:k
463:X
442:1
439:=
431:+
428:k
424:X
420:=
414:=
409:1
406:+
403:k
399:X
395:=
390:k
386:X
364:N
357:k
335:k
331:X
308:k
304:X
281:k
277:X
256:F
234:F
212:F
186:F
178:F
153:k
149:X
126:F
103:N
96:k
74:k
70:X
23:.
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