652:, since its occurrence is unchanged under transpositions (for a finite re-ordering, the convergence or divergence of the series—and, indeed, the numerical value of the sum itself—is independent of the order in which we add up the terms). Thus, the series either converges almost surely or diverges almost surely. If we assume in addition that the common
949: = 0 infinitely often } is invariant under finite permutations. Therefore, the zero–one law is applicable and one infers that the probability of a random walk with real iid increments visiting the origin infinitely often is either one or zero. Visiting the origin infinitely often is a tail event with respect to the sequence (
836:
i.e. the series diverges almost surely. This is a particularly simple application of the Hewitt–Savage zero–one law. In many situations, it can be easy to apply the Hewitt–Savage zero–one law to show that some event has probability 0 or 1, but surprisingly hard to determine
831:
363:
143:. The Hewitt-Savage zero–one law says that any event whose occurrence or non-occurrence is determined by the values of these random variables and whose occurrence or non-occurrence is unchanged by finite
546:
296:
236:
111:
914:
747:
626:
118:
480:
697:
650:
414:
183:
141:
578:
446:
390:
755:
301:
968:
29:
1053:
498:
248:
188:
63:
369:
855:
702:
583:
33:
1058:
155:
451:
992:
49:
658:
151:
either 0 or 1 (a "finite" permutation is one that leaves all but finitely many of the indices fixed).
631:
395:
164:
124:
551:
25:
1008:
419:
1029:
653:
375:
1013:
996:
416:), it is enough to check if its occurrence is unchanged by an arbitrary transposition
1047:
37:
988:
45:
924:
242:
148:
144:
826:{\displaystyle \mathbb {P} \left(\sum _{n=1}^{\infty }X_{n}=+\infty \right)=1,}
548:
of independent and identically distributed random variables take values in
114:
239:
749:
because of the random variables' non-negativity), we may conclude that
21:
358:{\displaystyle A\in {\mathcal {E}}\implies \mathbb {P} (A)\in \{0,1\}}
40:
happen or almost surely not happen. It is sometimes known as the
185:
to be the set of events (depending on the sequence of variables
1036:(Second ed.). New York: Springer-Verlag. pp. 381–82.
928:
637:
401:
368:
Since any finite permutation can be written as a product of
313:
170:
36:, that specifies that a certain type of event will either
119:
independent and identically-distributed random variables
628:
converges (to a finite value) is a symmetric event in
858:
758:
705:
661:
634:
586:
554:
501:
454:
422:
398:
378:
304:
251:
191:
167:
127:
66:
541:{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}
291:{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}
231:{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}
106:{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}
908:
825:
741:
691:
644:
620:
572:
540:
474:
440:
408:
384:
357:
290:
230:
177:
135:
105:
841:of these two extreme values is the correct one.
372:, if we wish to check whether or not an event
849:Continuing with the previous example, define
8:
352:
340:
909:{\displaystyle S_{N}=\sum _{n=1}^{N}X_{n},}
56:Statement of the Hewitt-Savage zero-one law
997:"Symmetric measures on Cartesian products"
742:{\displaystyle \mathbb {P} (X_{n}=0)<1}
621:{\displaystyle \sum _{n=1}^{\infty }X_{n}}
322:
318:
1012:
897:
887:
876:
863:
857:
794:
784:
773:
760:
759:
757:
718:
707:
706:
704:
674:
663:
662:
660:
636:
635:
633:
612:
602:
591:
585:
553:
532:
521:
511:
500:
468:
467:
453:
421:
400:
399:
397:
377:
324:
323:
312:
311:
303:
282:
271:
261:
250:
222:
211:
201:
190:
169:
168:
166:
129:
128:
126:
97:
86:
76:
65:
980:
967:are not independent and therefore the
42:Savage-Hewitt law for symmetric events
154:Somewhat more abstractly, define the
7:
475:{\displaystyle i,j\in \mathbb {N} }
806:
785:
603:
564:
533:
283:
223:
98:
14:
1014:10.1090/s0002-9947-1955-0076206-8
971:is not directly applicable here.
692:{\displaystyle \mathbb {E} >0}
580:. Then the event that the series
160:sigma algebra of symmetric events
245:of the indices in the sequence
919:which is the position at step
730:
711:
699:(which essentially means that
680:
667:
645:{\displaystyle {\mathcal {E}}}
567:
555:
435:
423:
409:{\displaystyle {\mathcal {E}}}
334:
328:
319:
178:{\displaystyle {\mathcal {E}}}
1:
238:) which are invariant under
136:{\displaystyle \mathbb {X} }
573:{\displaystyle [0,\infty )}
1075:
156:exchangeable sigma algebra
18:Hewitt–Savage zero–one law
969:Kolmogorov's zero–one law
30:Kolmogorov's zero–one law
121:taking values in a set
1001:Trans. Amer. Math. Soc
910:
892:
827:
789:
743:
693:
646:
622:
607:
574:
542:
476:
442:
410:
392:is symmetric (lies in
386:
359:
292:
232:
179:
137:
107:
1027:This example is from
911:
872:
828:
769:
744:
694:
647:
623:
587:
575:
543:
477:
443:
441:{\displaystyle (i,j)}
411:
387:
360:
293:
233:
180:
138:
108:
50:Leonard Jimmie Savage
1054:Probability theorems
856:
756:
703:
659:
632:
584:
552:
499:
452:
420:
396:
376:
302:
249:
189:
165:
147:of the indices, has
125:
64:
44:. It is named after
34:Borel–Cantelli lemma
940:. The event {
537:
287:
227:
102:
1034:Probability Theory
906:
823:
739:
689:
642:
618:
570:
538:
502:
472:
438:
406:
382:
355:
288:
252:
228:
192:
175:
133:
103:
67:
26:probability theory
495:Let the sequence
385:{\displaystyle A}
1066:
1038:
1037:
1025:
1019:
1018:
1016:
985:
915:
913:
912:
907:
902:
901:
891:
886:
868:
867:
832:
830:
829:
824:
813:
809:
799:
798:
788:
783:
763:
748:
746:
745:
740:
723:
722:
710:
698:
696:
695:
690:
679:
678:
666:
651:
649:
648:
643:
641:
640:
627:
625:
624:
619:
617:
616:
606:
601:
579:
577:
576:
571:
547:
545:
544:
539:
536:
531:
520:
516:
515:
481:
479:
478:
473:
471:
447:
445:
444:
439:
415:
413:
412:
407:
405:
404:
391:
389:
388:
383:
364:
362:
361:
356:
327:
317:
316:
297:
295:
294:
289:
286:
281:
270:
266:
265:
237:
235:
234:
229:
226:
221:
210:
206:
205:
184:
182:
181:
176:
174:
173:
142:
140:
139:
134:
132:
112:
110:
109:
104:
101:
96:
85:
81:
80:
1074:
1073:
1069:
1068:
1067:
1065:
1064:
1063:
1059:Covering lemmas
1044:
1043:
1042:
1041:
1028:
1026:
1022:
987:
986:
982:
977:
966:
957:
948:
939:
893:
859:
854:
853:
847:
790:
768:
764:
754:
753:
714:
701:
700:
670:
657:
656:
630:
629:
608:
582:
581:
550:
549:
507:
503:
497:
496:
493:
488:
450:
449:
418:
417:
394:
393:
374:
373:
300:
299:
257:
253:
247:
246:
197:
193:
187:
186:
163:
162:
123:
122:
72:
68:
62:
61:
58:
12:
11:
5:
1072:
1070:
1062:
1061:
1056:
1046:
1045:
1040:
1039:
1020:
979:
978:
976:
973:
962:
953:
944:
935:
917:
916:
905:
900:
896:
890:
885:
882:
879:
875:
871:
866:
862:
846:
843:
834:
833:
822:
819:
816:
812:
808:
805:
802:
797:
793:
787:
782:
779:
776:
772:
767:
762:
738:
735:
732:
729:
726:
721:
717:
713:
709:
688:
685:
682:
677:
673:
669:
665:
654:expected value
639:
615:
611:
605:
600:
597:
594:
590:
569:
566:
563:
560:
557:
535:
530:
527:
524:
519:
514:
510:
506:
492:
489:
487:
484:
470:
466:
463:
460:
457:
437:
434:
431:
428:
425:
403:
381:
370:transpositions
354:
351:
348:
345:
342:
339:
336:
333:
330:
326:
321:
315:
310:
307:
285:
280:
277:
274:
269:
264:
260:
256:
225:
220:
217:
214:
209:
204:
200:
196:
172:
131:
100:
95:
92:
89:
84:
79:
75:
71:
57:
54:
13:
10:
9:
6:
4:
3:
2:
1071:
1060:
1057:
1055:
1052:
1051:
1049:
1035:
1031:
1024:
1021:
1015:
1010:
1006:
1002:
998:
994:
993:Savage, L. J.
990:
984:
981:
974:
972:
970:
965:
961:
956:
952:
947:
943:
938:
934:
930:
926:
922:
903:
898:
894:
888:
883:
880:
877:
873:
869:
864:
860:
852:
851:
850:
844:
842:
840:
820:
817:
814:
810:
803:
800:
795:
791:
780:
777:
774:
770:
765:
752:
751:
750:
736:
733:
727:
724:
719:
715:
686:
683:
675:
671:
655:
613:
609:
598:
595:
592:
588:
561:
558:
528:
525:
522:
517:
512:
508:
504:
490:
485:
483:
464:
461:
458:
455:
432:
429:
426:
379:
371:
366:
349:
346:
343:
337:
331:
308:
305:
278:
275:
272:
267:
262:
258:
254:
244:
241:
218:
215:
212:
207:
202:
198:
194:
161:
157:
152:
150:
146:
120:
116:
93:
90:
87:
82:
77:
73:
69:
55:
53:
51:
47:
43:
39:
38:almost surely
35:
31:
28:, similar to
27:
23:
19:
1033:
1030:Shiryaev, A.
1023:
1004:
1000:
983:
963:
959:
954:
950:
945:
941:
936:
932:
920:
918:
848:
838:
835:
494:
367:
243:permutations
159:
153:
145:permutations
59:
46:Edwin Hewitt
41:
17:
15:
1007:: 470–501.
931:increments
925:random walk
149:probability
1048:Categories
989:Hewitt, E.
975:References
927:with the
874:∑
845:Example 2
807:∞
786:∞
771:∑
604:∞
589:∑
565:∞
534:∞
491:Example 1
465:∈
338:∈
320:⟹
309:∈
284:∞
224:∞
99:∞
1032:(1996).
995:(1955).
486:Examples
115:sequence
32:and the
958:), but
298:. Then
22:theorem
240:finite
923:of a
839:which
113:be a
20:is a
734:<
684:>
60:Let
48:and
16:The
1009:doi
929:iid
158:or
117:of
24:in
1050::
1005:80
1003:.
999:.
991:;
482:.
448:,
365:.
52:.
1017:.
1011::
964:N
960:S
955:N
951:S
946:N
942:S
937:n
933:X
921:N
904:,
899:n
895:X
889:N
884:1
881:=
878:n
870:=
865:N
861:S
821:,
818:1
815:=
811:)
804:+
801:=
796:n
792:X
781:1
778:=
775:n
766:(
761:P
737:1
731:)
728:0
725:=
720:n
716:X
712:(
708:P
687:0
681:]
676:n
672:X
668:[
664:E
638:E
614:n
610:X
599:1
596:=
593:n
568:)
562:,
559:0
556:[
529:1
526:=
523:n
518:}
513:n
509:X
505:{
469:N
462:j
459:,
456:i
436:)
433:j
430:,
427:i
424:(
402:E
380:A
353:}
350:1
347:,
344:0
341:{
335:)
332:A
329:(
325:P
314:E
306:A
279:1
276:=
273:n
268:}
263:n
259:X
255:{
219:1
216:=
213:n
208:}
203:n
199:X
195:{
171:E
130:X
94:1
91:=
88:n
83:}
78:n
74:X
70:{
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