990:
542:
to the sphere). The derivative is random, with zero expectation and some standard deviation. The latter may depend on the point and the direction. However, if it does not depend, then it is equal to
1423:
The basic statement given above is a simple special case of a much more general (and difficult) theory stated by Adler. For a detailed presentation of this special case see
Tsirelson's lectures.
342:
607:
1356:
885:
685:
1103:
1243:
1161:
1040:
794:
1391:
1274:
849:
759:
262:
1320:
1211:
1129:
1070:
397:
227:
362:
1294:
1185:
814:
716:
650:
627:
536:
512:
489:
465:
441:
421:
201:
181:
161:
141:
121:
101:
893:
1456:
49:
1163:
is non-empty; its Euler characteristic may take various values, depending on the topology of the set (the number of
267:
37:
33:
75:. Though rare in a small domain (of space or/and time), large deviations may be quite usual in a large domain.
1498:
545:
400:
56:
996:
719:
1000:
1325:
854:
655:
1412:
1075:
1216:
1134:
1013:
767:
1361:
1252:
819:
737:
515:
232:
1407:
1480:
1299:
1190:
1108:
1049:
367:
206:
688:
444:
347:
8:
1279:
1249:(which is easy to guess, but quite difficult to prove). Thus, its Euler characteristic
1170:
799:
701:
635:
612:
521:
497:
474:
450:
426:
406:
186:
166:
146:
126:
106:
86:
68:
1452:
1004:
762:
1402:
1451:
Robert J. Adler, Jonathan E. Taylor, "Random fields and geometry", Springer 2007.
1438:
Robert J. Adler, "On excursion sets, tube formulas and maxima of random fields",
1164:
468:
64:
21:
1492:
1439:
723:
514:
is easy to determine in the important special case described in terms of the
25:
539:
1467:
Robert J. Adler, "Some new random field tools for spatial analysis",
1043:
123:
on the (two-dimensional) sphere. Assume that the expected value of
63:
Sometimes, a value of a
Gaussian random function deviates from its
1468:
1440:
The Annals of
Applied Probability 2000, Vol. 10, No. 1, 1–74
1246:
734:
The clue to the theory sketched above is, Euler characteristic
41:
726:), and reaches its maximum at a single point (almost surely).
163:(at every point of the sphere), and the standard deviation of
692:
985:{\displaystyle E(\chi _{a})=Ca\exp(-a^{2}/2)+2P(\xi >a)}
1167:, and possible holes in these components). However, if
538:
at a given point (of the sphere) in a given direction (
729:
1364:
1328:
1302:
1282:
1255:
1219:
1193:
1173:
1137:
1111:
1078:
1052:
1016:
896:
857:
822:
802:
770:
740:
704:
658:
638:
615:
548:
524:
500:
477:
453:
429:
409:
370:
350:
270:
235:
209:
189:
169:
149:
129:
109:
89:
471:
of the approximation decays exponentially for large
103:be the maximal value of a Gaussian random function
1385:
1350:
1314:
1288:
1268:
1237:
1205:
1179:
1155:
1123:
1097:
1064:
1034:
984:
879:
851:. Its expected value (in other words, mean value)
843:
808:
788:
753:
710:
679:
644:
621:
601:
530:
506:
483:
459:
435:
415:
391:
356:
336:
256:
221:
195:
175:
155:
135:
115:
95:
1490:
203:(at every point of the sphere). Then, for large
995:(which is far from being trivial, and involves
1245:is usually a small, slightly deformed disk or
337:{\displaystyle Ca\exp(-a^{2}/2)+2P(\xi >a)}
1232:
1220:
1150:
1138:
1029:
1017:
783:
771:
1434:
1432:
1430:
1461:
1427:
44:are useful (for example) when analysing
1491:
1445:
602:{\displaystyle (\pi /2)^{1/4}C^{1/2}}
423:is a constant; it does not depend on
50:cosmic microwave background radiation
730:The clue: mean Euler characteristic
13:
1418:
78:
20:– of either one variable (a
14:
1510:
40:. Gaussian random fields on the
38:multivariate normal distribution
34:finite-dimensional distribution
24:), or two or more variables (a
1474:
1380:
1368:
1345:
1332:
979:
967:
955:
931:
913:
900:
887:can be calculated explicitly:
874:
861:
832:
826:
674:
662:
564:
549:
386:
374:
331:
319:
307:
283:
251:
239:
1:
1351:{\displaystyle E(\chi _{a})}
880:{\displaystyle E(\chi _{a})}
680:{\displaystyle P(\xi >a)}
401:standard normal distribution
57:positron emission tomography
7:
1396:
1098:{\displaystyle \chi _{a}=0}
720:continuously differentiable
59:(see, pp. 9–10).
10:
1515:
1442:. (Special invited paper.)
1238:{\displaystyle \{X>a\}}
1156:{\displaystyle \{X>a\}}
1105:. In the other case, when
1035:{\displaystyle \{X>a\}}
816:(of the sphere) such that
789:{\displaystyle \{X>a\}}
609:(for the sphere of radius
52:(see, pp. 8–9);
1386:{\displaystyle P(M>a)}
1269:{\displaystyle \chi _{a}}
844:{\displaystyle X(t)>a}
754:{\displaystyle \chi _{a}}
257:{\displaystyle P(M>a)}
55:brain images obtained by
1481:Lectures of B. Tsirelson
1413:Large deviations theory
691:of the sphere (for the
1483:(especially, Sect. 5).
1387:
1352:
1316:
1315:{\displaystyle M>a}
1290:
1270:
1239:
1207:
1206:{\displaystyle M>a}
1181:
1157:
1125:
1124:{\displaystyle M>a}
1099:
1066:
1065:{\displaystyle M<a}
1036:
986:
881:
845:
810:
790:
755:
712:
681:
646:
623:
603:
532:
516:directional derivative
508:
485:
461:
437:
417:
393:
392:{\displaystyle N(0,1)}
358:
338:
258:
223:
222:{\displaystyle a>0}
197:
177:
157:
137:
117:
97:
1408:Gaussian random field
1388:
1353:
1317:
1291:
1271:
1240:
1208:
1182:
1158:
1126:
1100:
1067:
1037:
997:Poincaré–Hopf theorem
987:
882:
846:
811:
791:
756:
713:
682:
647:
624:
604:
533:
509:
486:
462:
443:, but depends on the
438:
418:
394:
359:
339:
259:
224:
198:
178:
158:
138:
118:
98:
48:the anomalies in the
1499:Stochastic processes
1362:
1326:
1300:
1280:
1276:is usually equal to
1253:
1217:
1191:
1171:
1165:connected components
1135:
1109:
1076:
1050:
1014:
1001:Gauss–Bonnet theorem
894:
855:
820:
800:
768:
738:
702:
689:Euler characteristic
656:
636:
613:
546:
522:
498:
475:
451:
445:correlation function
427:
407:
368:
357:{\displaystyle \xi }
348:
268:
233:
207:
187:
167:
147:
127:
107:
87:
28:) – is called
698:It is assumed that
69:standard deviations
1383:
1348:
1312:
1286:
1266:
1235:
1203:
1177:
1153:
1121:
1095:
1062:
1032:
982:
877:
841:
806:
786:
751:
708:
677:
642:
619:
599:
528:
504:
481:
457:
433:
413:
389:
354:
334:
254:
219:
193:
173:
153:
133:
113:
93:
1457:978-0-387-48112-8
1289:{\displaystyle 1}
1180:{\displaystyle a}
809:{\displaystyle t}
711:{\displaystyle X}
645:{\displaystyle 2}
622:{\displaystyle 1}
531:{\displaystyle X}
507:{\displaystyle C}
484:{\displaystyle a}
467:(see below). The
460:{\displaystyle X}
436:{\displaystyle a}
416:{\displaystyle C}
196:{\displaystyle 1}
176:{\displaystyle X}
156:{\displaystyle 0}
136:{\displaystyle X}
116:{\displaystyle X}
96:{\displaystyle M}
1506:
1484:
1478:
1472:
1465:
1459:
1449:
1443:
1436:
1403:Gaussian process
1392:
1390:
1389:
1384:
1357:
1355:
1354:
1349:
1344:
1343:
1321:
1319:
1318:
1313:
1295:
1293:
1292:
1287:
1275:
1273:
1272:
1267:
1265:
1264:
1244:
1242:
1241:
1236:
1212:
1210:
1209:
1204:
1186:
1184:
1183:
1178:
1162:
1160:
1159:
1154:
1130:
1128:
1127:
1122:
1104:
1102:
1101:
1096:
1088:
1087:
1071:
1069:
1068:
1063:
1041:
1039:
1038:
1033:
991:
989:
988:
983:
951:
946:
945:
912:
911:
886:
884:
883:
878:
873:
872:
850:
848:
847:
842:
815:
813:
812:
807:
795:
793:
792:
787:
760:
758:
757:
752:
750:
749:
717:
715:
714:
709:
686:
684:
683:
678:
651:
649:
648:
643:
632:The coefficient
628:
626:
625:
620:
608:
606:
605:
600:
598:
597:
593:
580:
579:
575:
559:
537:
535:
534:
529:
513:
511:
510:
505:
490:
488:
487:
482:
466:
464:
463:
458:
442:
440:
439:
434:
422:
420:
419:
414:
398:
396:
395:
390:
363:
361:
360:
355:
343:
341:
340:
335:
303:
298:
297:
263:
261:
260:
255:
228:
226:
225:
220:
202:
200:
199:
194:
182:
180:
179:
174:
162:
160:
159:
154:
142:
140:
139:
134:
122:
120:
119:
114:
102:
100:
99:
94:
1514:
1513:
1509:
1508:
1507:
1505:
1504:
1503:
1489:
1488:
1487:
1479:
1475:
1469:arXiv:0805.1031
1466:
1462:
1450:
1446:
1437:
1428:
1421:
1419:Further reading
1399:
1363:
1360:
1359:
1339:
1335:
1327:
1324:
1323:
1322:). This is why
1301:
1298:
1297:
1281:
1278:
1277:
1260:
1256:
1254:
1251:
1250:
1218:
1215:
1214:
1192:
1189:
1188:
1172:
1169:
1168:
1136:
1133:
1132:
1110:
1107:
1106:
1083:
1079:
1077:
1074:
1073:
1072:; in this case
1051:
1048:
1047:
1015:
1012:
1011:
947:
941:
937:
907:
903:
895:
892:
891:
868:
864:
856:
853:
852:
821:
818:
817:
801:
798:
797:
769:
766:
765:
745:
741:
739:
736:
735:
732:
703:
700:
699:
687:is in fact the
657:
654:
653:
637:
634:
633:
614:
611:
610:
589:
585:
581:
571:
567:
563:
555:
547:
544:
543:
523:
520:
519:
499:
496:
495:
476:
473:
472:
452:
449:
448:
428:
425:
424:
408:
405:
404:
369:
366:
365:
364:is distributed
349:
346:
345:
299:
293:
289:
269:
266:
265:
234:
231:
230:
208:
205:
204:
188:
185:
184:
168:
165:
164:
148:
145:
144:
128:
125:
124:
108:
105:
104:
88:
85:
84:
81:
79:Basic statement
73:large deviation
18:random function
12:
11:
5:
1512:
1502:
1501:
1486:
1485:
1473:
1460:
1444:
1425:
1420:
1417:
1416:
1415:
1410:
1405:
1398:
1395:
1382:
1379:
1376:
1373:
1370:
1367:
1347:
1342:
1338:
1334:
1331:
1311:
1308:
1305:
1285:
1263:
1259:
1234:
1231:
1228:
1225:
1222:
1202:
1199:
1196:
1176:
1152:
1149:
1146:
1143:
1140:
1120:
1117:
1114:
1094:
1091:
1086:
1082:
1061:
1058:
1055:
1031:
1028:
1025:
1022:
1019:
1005:Rice's formula
993:
992:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
950:
944:
940:
936:
933:
930:
927:
924:
921:
918:
915:
910:
906:
902:
899:
876:
871:
867:
863:
860:
840:
837:
834:
831:
828:
825:
805:
796:of all points
785:
782:
779:
776:
773:
748:
744:
731:
728:
707:
695:it vanishes).
676:
673:
670:
667:
664:
661:
641:
618:
596:
592:
588:
584:
578:
574:
570:
566:
562:
558:
554:
551:
527:
503:
480:
469:relative error
456:
432:
412:
388:
385:
382:
379:
376:
373:
353:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
302:
296:
292:
288:
285:
282:
279:
276:
273:
253:
250:
247:
244:
241:
238:
218:
215:
212:
192:
172:
152:
132:
112:
92:
80:
77:
65:expected value
61:
60:
53:
22:random process
9:
6:
4:
3:
2:
1511:
1500:
1497:
1496:
1494:
1482:
1477:
1470:
1464:
1458:
1454:
1448:
1441:
1435:
1433:
1431:
1426:
1424:
1414:
1411:
1409:
1406:
1404:
1401:
1400:
1394:
1377:
1374:
1371:
1365:
1340:
1336:
1329:
1309:
1306:
1303:
1283:
1261:
1257:
1248:
1229:
1226:
1223:
1213:then the set
1200:
1197:
1194:
1187:is large and
1174:
1166:
1147:
1144:
1141:
1118:
1115:
1112:
1092:
1089:
1084:
1080:
1059:
1056:
1053:
1045:
1026:
1023:
1020:
1008:
1006:
1002:
998:
976:
973:
970:
964:
961:
958:
952:
948:
942:
938:
934:
928:
925:
922:
919:
916:
908:
904:
897:
890:
889:
888:
869:
865:
858:
838:
835:
829:
823:
803:
780:
777:
774:
764:
746:
742:
727:
725:
724:almost surely
721:
705:
696:
694:
690:
671:
668:
665:
659:
639:
630:
616:
594:
590:
586:
582:
576:
572:
568:
560:
556:
552:
541:
525:
517:
501:
494:The constant
492:
478:
470:
454:
446:
430:
410:
402:
383:
380:
377:
371:
351:
328:
325:
322:
316:
313:
310:
304:
300:
294:
290:
286:
280:
277:
274:
271:
248:
245:
242:
236:
216:
213:
210:
190:
170:
150:
130:
110:
90:
76:
74:
70:
66:
58:
54:
51:
47:
46:
45:
43:
39:
35:
31:
27:
23:
19:
1476:
1463:
1447:
1422:
1358:is close to
1296:(given that
1009:
994:
733:
697:
631:
493:
264:is close to
82:
72:
71:. This is a
62:
29:
26:random field
17:
15:
1131:, the set
67:by several
540:tangential
1337:χ
1258:χ
1081:χ
1046:whenever
1044:empty set
971:ξ
935:−
929:
905:χ
866:χ
743:χ
718:is twice
666:ξ
553:π
352:ξ
323:ξ
287:−
281:
32:if every
1493:Category
1397:See also
1010:The set
344:, where
30:Gaussian
1247:ellipse
1042:is the
1007:etc.).
761:of the
652:before
403:), and
1455:
42:sphere
693:torus
399:(the
36:is a
1453:ISBN
1375:>
1307:>
1227:>
1198:>
1145:>
1116:>
1057:<
1024:>
974:>
836:>
778:>
669:>
326:>
246:>
214:>
83:Let
926:exp
763:set
629:).
518:of
447:of
278:exp
183:is
143:is
1495::
1429:^
1393:.
1003:,
999:,
491:.
229:,
16:A
1471:.
1381:)
1378:a
1372:M
1369:(
1366:P
1346:)
1341:a
1333:(
1330:E
1310:a
1304:M
1284:1
1262:a
1233:}
1230:a
1224:X
1221:{
1201:a
1195:M
1175:a
1151:}
1148:a
1142:X
1139:{
1119:a
1113:M
1093:0
1090:=
1085:a
1060:a
1054:M
1030:}
1027:a
1021:X
1018:{
980:)
977:a
968:(
965:P
962:2
959:+
956:)
953:2
949:/
943:2
939:a
932:(
923:a
920:C
917:=
914:)
909:a
901:(
898:E
875:)
870:a
862:(
859:E
839:a
833:)
830:t
827:(
824:X
804:t
784:}
781:a
775:X
772:{
747:a
722:(
706:X
675:)
672:a
663:(
660:P
640:2
617:1
595:2
591:/
587:1
583:C
577:4
573:/
569:1
565:)
561:2
557:/
550:(
526:X
502:C
479:a
455:X
431:a
411:C
387:)
384:1
381:,
378:0
375:(
372:N
332:)
329:a
320:(
317:P
314:2
311:+
308:)
305:2
301:/
295:2
291:a
284:(
275:a
272:C
252:)
249:a
243:M
240:(
237:P
217:0
211:a
191:1
171:X
151:0
131:X
111:X
91:M
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