710:
536:
528:
422:
840:
C. B. Mehr and J. A. McFadden. Certain
Properties of Gaussian Processes and Their First-Passage Times. Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 3(1965), pp. 505-522
299:
705:{\displaystyle {\textbf {S}}_{x}(s)={\frac {2\sigma ^{2}\beta }{-s^{2}+\beta ^{2}}}={\frac {{\sqrt {2\beta }}\,\sigma }{(s+\beta )}}\cdot {\frac {{\sqrt {2\beta }}\,\sigma }{(-s+\beta )}}.}
913:
437:
1448:
332:
1272:
827:
1875:
868:
1405:
1385:
1789:
345:
801:
1706:
1716:
1390:
222:
Property (3) means that every non-degenerate mean-square continuous Gauss–Markov process can be synthesized from the standard Wiener process (SWP).
1400:
1758:
1473:
1655:
1945:
1935:
1458:
811:
784:
1845:
1809:
1762:
2113:
1850:
960:
861:
1915:
1493:
1463:
1766:
1750:
1960:
1665:
885:
1865:
1830:
1799:
1794:
1433:
1230:
1147:
1804:
1132:
231:
58:
1428:
1235:
751:
1154:
1890:
1770:
2118:
1895:
1731:
1630:
1615:
1027:
943:
854:
1905:
1541:
242:
1900:
1503:
1087:
1032:
948:
1835:
1825:
1468:
1438:
1840:
1005:
903:
1551:
1127:
908:
2139:
1920:
1721:
1635:
1620:
1010:
1754:
1062:
28:
1142:
1117:
1860:
1443:
978:
2055:
2045:
1736:
1518:
1257:
1122:
933:
1340:
1997:
1925:
1184:
162:
If the process is non-degenerate and mean-square continuous, then there exists a non-zero scalar function
2020:
2002:
1982:
1977:
1696:
1528:
1508:
1355:
1298:
1137:
1047:
1488:
2095:
2050:
2040:
1781:
1726:
1701:
1670:
1650:
1410:
1395:
1262:
38:
2090:
1930:
1855:
1660:
1420:
1330:
1220:
431:
57:. A stationary Gauss–Markov process is unique up to rescaling; such a process is also known as an
2060:
2025:
1940:
1910:
1680:
1675:
1498:
1335:
1000:
938:
877:
821:
523:{\displaystyle {\textbf {S}}_{x}(j\omega )={\frac {2\sigma ^{2}\beta }{\omega ^{2}+\beta ^{2}}}.}
307:
46:
1741:
2080:
1885:
1536:
1293:
1210:
1179:
1072:
1052:
1042:
898:
893:
807:
780:
776:
747:
65:
1746:
1483:
2100:
1987:
1870:
1240:
1215:
1164:
1092:
1015:
968:
768:
427:
50:
2065:
1965:
1950:
1711:
1645:
1323:
1267:
1250:
995:
339:
1880:
1112:
2070:
2035:
1955:
1561:
1308:
1225:
1194:
1189:
1169:
1159:
1102:
1097:
1077:
1057:
1022:
990:
973:
738:
215:
54:
2133:
1972:
1513:
1350:
1345:
1303:
1245:
1067:
983:
923:
769:
713:
302:
42:
17:
2030:
1992:
1546:
1478:
1367:
1362:
1174:
1107:
1082:
918:
1610:
2075:
1594:
1589:
1584:
1574:
1377:
1318:
1313:
1277:
1037:
928:
2085:
1625:
1569:
1453:
530:(Note that the Cauchy distribution and this spectrum differ by scale factors.)
1579:
237:
1406:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
846:
417:{\displaystyle {\textbf {R}}_{x}(\tau )=\sigma ^{2}e^{-\beta |\tau |}.}
850:
720:
There are also some trivial exceptions to all of the above.
1386:
Autoregressive conditional heteroskedasticity (ARCH) model
232:
Ornstein–Uhlenbeck process § Mathematical properties
914:
Independent and identically distributed random variables
800:
Bob Schutz, Byron Tapley, George H. Born (2004-06-26).
771:
3D-Position
Tracking and Control for All-Terrain Robots
1391:
Autoregressive integrated moving average (ARIMA) model
533:
The above yields the following spectral factorization:
539:
440:
348:
310:
245:
2013:
1818:
1780:
1689:
1603:
1560:
1527:
1419:
1376:
1286:
1203:
959:
884:
294:{\displaystyle {\textbf {E}}(X^{2}(t))=\sigma ^{2}}
704:
522:
416:
326:
293:
1273:Stochastic chains with memory of variable length
737:C. E. Rasmussen & C. K. I. Williams (2006).
430:(PSD) function that has the same shape as the
862:
8:
826:: CS1 maint: multiple names: authors list (
170:) and a strictly increasing scalar function
84:) possesses the three following properties:
1401:Autoregressive–moving-average (ARMA) model
869:
855:
847:
672:
662:
659:
632:
622:
619:
607:
594:
576:
566:
548:
542:
541:
538:
508:
495:
480:
470:
449:
443:
442:
439:
404:
396:
389:
379:
357:
351:
350:
347:
315:
309:
285:
260:
247:
246:
244:
135:) is a non-decreasing scalar function of
740:Gaussian Processes for Machine Learning
729:
236:A stationary Gauss–Markov process with
49:that satisfy the requirements for both
1707:Doob's martingale convergence theorems
819:
1459:Constant elasticity of variance (CEV)
1449:Chan–Karolyi–Longstaff–Sanders (CKLS)
7:
543:
444:
352:
248:
96:) is a non-zero scalar function of
1946:Skorokhod's representation theorem
1727:Law of large numbers (weak/strong)
25:
1916:Martingale representation theorem
746:. MIT Press. p. Appendix B.
159:)) is also a Gauss–Markov process
35:Gauss–Markov stochastic processes
1961:Stochastic differential equation
1851:Doob's optional stopping theorem
1846:Doob–Meyer decomposition theorem
124:) is also a Gauss–Markov process
1831:Convergence of random variables
1717:Fisher–Tippett–Gnedenko theorem
803:Statistical Orbit Determination
1429:Binomial options pricing model
693:
678:
650:
638:
560:
554:
464:
455:
405:
397:
369:
363:
334:has the following properties.
275:
272:
266:
253:
1:
1896:Kolmogorov continuity theorem
1732:Law of the iterated logarithm
1901:Kolmogorov extension theorem
1580:Generalized queueing network
1088:Interacting particle systems
64:Gauss–Markov processes obey
27:Not to be confused with the
1033:Continuous-time random walk
327:{\displaystyle \beta ^{-1}}
76:Every Gauss–Markov process
31:of mathematical statistics.
2156:
2041:Extreme value theory (EVT)
1841:Doob decomposition theorem
1133:Ornstein–Uhlenbeck process
904:Chinese restaurant process
229:
59:Ornstein–Uhlenbeck process
26:
2109:
1921:Optional stopping theorem
1722:Large deviation principle
1474:Heath–Jarrow–Morton (HJM)
1411:Moving-average (MA) model
1396:Autoregressive (AR) model
1221:Hidden Markov model (HMM)
1155:Schramm–Loewner evolution
1836:Doléans-Dade exponential
1666:Progressively measurable
1464:Cox–Ingersoll–Ross (CIR)
2056:Mathematical statistics
2046:Large deviations theory
1876:Infinitesimal generator
1737:Maximal ergodic theorem
1656:Piecewise-deterministic
1258:Random dynamical system
1123:Markov additive process
1891:Karhunen–Loève theorem
1826:Cameron–Martin formula
1790:Burkholder–Davis–Gundy
1185:Variance gamma process
767:Lamon, Pierre (2008).
712:which is important in
706:
524:
418:
328:
295:
2021:Actuarial mathematics
1983:Uniform integrability
1978:Stratonovich integral
1906:Lévy–Prokhorov metric
1810:Marcinkiewicz–Zygmund
1697:Central limit theorem
1299:Gaussian random field
1128:McKean–Vlasov process
1048:Dyson Brownian motion
909:Galton–Watson process
775:. Springer. pp.
707:
525:
419:
329:
296:
2096:Time series analysis
2051:Mathematical finance
1936:Reflection principle
1263:Regenerative process
1063:Fleming–Viot process
878:Stochastic processes
537:
438:
346:
308:
243:
47:stochastic processes
39:Carl Friedrich Gauss
29:Gauss–Markov theorem
18:Gauss-Markov process
2091:Stochastic analysis
1931:Quadratic variation
1926:Prokhorov's theorem
1861:Feynman–Kac formula
1331:Markov random field
979:Birth–death process
432:Cauchy distribution
2061:Probability theory
1941:Skorokhod integral
1911:Malliavin calculus
1494:Korn-Kreer-Lenssen
1378:Time series models
1341:Pitman–Yor process
702:
520:
414:
324:
291:
214:) is the standard
66:Langevin equations
51:Gaussian processes
2127:
2126:
2081:Signal processing
1800:Doob's upcrossing
1795:Doob's martingale
1759:Engelbert–Schmidt
1702:Donsker's theorem
1636:Feller-continuous
1504:Rendleman–Bartter
1294:Dirichlet process
1211:Branching process
1180:Telegraph process
1073:Geometric process
1053:Empirical process
1043:Diffusion process
899:Branching process
894:Bernoulli process
813:978-0-08-054173-0
786:978-3-540-78286-5
697:
670:
654:
630:
614:
545:
515:
446:
354:
250:
16:(Redirected from
2147:
2140:Markov processes
2101:Machine learning
1988:Usual hypotheses
1871:Girsanov theorem
1856:Dynkin's formula
1621:Continuous paths
1529:Actuarial models
1469:Garman–Kohlhagen
1439:Black–Karasinski
1434:Black–Derman–Toy
1421:Financial models
1287:Fields and other
1216:Gaussian process
1165:Sigma-martingale
969:Additive process
871:
864:
857:
848:
841:
838:
832:
831:
825:
817:
797:
791:
790:
774:
764:
758:
757:
745:
734:
716:and other areas.
714:Wiener filtering
711:
709:
708:
703:
698:
696:
676:
671:
663:
660:
655:
653:
636:
631:
623:
620:
615:
613:
612:
611:
599:
598:
585:
581:
580:
567:
553:
552:
547:
546:
529:
527:
526:
521:
516:
514:
513:
512:
500:
499:
489:
485:
484:
471:
454:
453:
448:
447:
428:spectral density
423:
421:
420:
415:
410:
409:
408:
400:
384:
383:
362:
361:
356:
355:
333:
331:
330:
325:
323:
322:
300:
298:
297:
292:
290:
289:
265:
264:
252:
251:
226:Other properties
72:Basic properties
55:Markov processes
21:
2155:
2154:
2150:
2149:
2148:
2146:
2145:
2144:
2130:
2129:
2128:
2123:
2105:
2066:Queueing theory
2009:
1951:Skorokhod space
1814:
1805:Kunita–Watanabe
1776:
1742:Sanov's theorem
1712:Ergodic theorem
1685:
1681:Time-reversible
1599:
1562:Queueing models
1556:
1552:Sparre–Anderson
1542:Cramér–Lundberg
1523:
1509:SABR volatility
1415:
1372:
1324:Boolean network
1282:
1268:Renewal process
1199:
1148:Non-homogeneous
1138:Poisson process
1028:Contact process
991:Brownian motion
961:Continuous time
955:
949:Maximal entropy
880:
875:
845:
844:
839:
835:
818:
814:
806:. p. 230.
799:
798:
794:
787:
766:
765:
761:
754:
743:
736:
735:
731:
726:
677:
661:
637:
621:
603:
590:
586:
572:
568:
540:
535:
534:
504:
491:
490:
476:
472:
441:
436:
435:
385:
375:
349:
344:
343:
340:autocorrelation
311:
306:
305:
281:
256:
241:
240:
234:
228:
74:
32:
23:
22:
15:
12:
11:
5:
2153:
2151:
2143:
2142:
2132:
2131:
2125:
2124:
2122:
2121:
2116:
2114:List of topics
2110:
2107:
2106:
2104:
2103:
2098:
2093:
2088:
2083:
2078:
2073:
2071:Renewal theory
2068:
2063:
2058:
2053:
2048:
2043:
2038:
2036:Ergodic theory
2033:
2028:
2026:Control theory
2023:
2017:
2015:
2011:
2010:
2008:
2007:
2006:
2005:
2000:
1990:
1985:
1980:
1975:
1970:
1969:
1968:
1958:
1956:Snell envelope
1953:
1948:
1943:
1938:
1933:
1928:
1923:
1918:
1913:
1908:
1903:
1898:
1893:
1888:
1883:
1878:
1873:
1868:
1863:
1858:
1853:
1848:
1843:
1838:
1833:
1828:
1822:
1820:
1816:
1815:
1813:
1812:
1807:
1802:
1797:
1792:
1786:
1784:
1778:
1777:
1775:
1774:
1755:Borel–Cantelli
1744:
1739:
1734:
1729:
1724:
1719:
1714:
1709:
1704:
1699:
1693:
1691:
1690:Limit theorems
1687:
1686:
1684:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1648:
1643:
1638:
1633:
1628:
1623:
1618:
1613:
1607:
1605:
1601:
1600:
1598:
1597:
1592:
1587:
1582:
1577:
1572:
1566:
1564:
1558:
1557:
1555:
1554:
1549:
1544:
1539:
1533:
1531:
1525:
1524:
1522:
1521:
1516:
1511:
1506:
1501:
1496:
1491:
1486:
1481:
1476:
1471:
1466:
1461:
1456:
1451:
1446:
1441:
1436:
1431:
1425:
1423:
1417:
1416:
1414:
1413:
1408:
1403:
1398:
1393:
1388:
1382:
1380:
1374:
1373:
1371:
1370:
1365:
1360:
1359:
1358:
1353:
1343:
1338:
1333:
1328:
1327:
1326:
1321:
1311:
1309:Hopfield model
1306:
1301:
1296:
1290:
1288:
1284:
1283:
1281:
1280:
1275:
1270:
1265:
1260:
1255:
1254:
1253:
1248:
1243:
1238:
1228:
1226:Markov process
1223:
1218:
1213:
1207:
1205:
1201:
1200:
1198:
1197:
1195:Wiener sausage
1192:
1190:Wiener process
1187:
1182:
1177:
1172:
1170:Stable process
1167:
1162:
1160:Semimartingale
1157:
1152:
1151:
1150:
1145:
1135:
1130:
1125:
1120:
1115:
1110:
1105:
1103:Jump diffusion
1100:
1095:
1090:
1085:
1080:
1078:Hawkes process
1075:
1070:
1065:
1060:
1058:Feller process
1055:
1050:
1045:
1040:
1035:
1030:
1025:
1023:Cauchy process
1020:
1019:
1018:
1013:
1008:
1003:
998:
988:
987:
986:
976:
974:Bessel process
971:
965:
963:
957:
956:
954:
953:
952:
951:
946:
941:
936:
926:
921:
916:
911:
906:
901:
896:
890:
888:
882:
881:
876:
874:
873:
866:
859:
851:
843:
842:
833:
812:
792:
785:
759:
752:
728:
727:
725:
722:
718:
717:
701:
695:
692:
689:
686:
683:
680:
675:
669:
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658:
652:
649:
646:
643:
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635:
629:
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618:
610:
606:
602:
597:
593:
589:
584:
579:
575:
571:
565:
562:
559:
556:
551:
531:
519:
511:
507:
503:
498:
494:
488:
483:
479:
475:
469:
466:
463:
460:
457:
452:
424:
413:
407:
403:
399:
395:
392:
388:
382:
378:
374:
371:
368:
365:
360:
321:
318:
314:
288:
284:
280:
277:
274:
271:
268:
263:
259:
255:
230:Main article:
227:
224:
220:
219:
216:Wiener process
160:
125:
73:
70:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2152:
2141:
2138:
2137:
2135:
2120:
2117:
2115:
2112:
2111:
2108:
2102:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2029:
2027:
2024:
2022:
2019:
2018:
2016:
2012:
2004:
2001:
1999:
1996:
1995:
1994:
1991:
1989:
1986:
1984:
1981:
1979:
1976:
1974:
1973:Stopping time
1971:
1967:
1964:
1963:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1917:
1914:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1823:
1821:
1817:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1787:
1785:
1783:
1779:
1772:
1768:
1764:
1763:Hewitt–Savage
1760:
1756:
1752:
1748:
1747:Zero–one laws
1745:
1743:
1740:
1738:
1735:
1733:
1730:
1728:
1725:
1723:
1720:
1718:
1715:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1694:
1692:
1688:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1632:
1629:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1608:
1606:
1602:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1567:
1565:
1563:
1559:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1535:
1534:
1532:
1530:
1526:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1444:Black–Scholes
1442:
1440:
1437:
1435:
1432:
1430:
1427:
1426:
1424:
1422:
1418:
1412:
1409:
1407:
1404:
1402:
1399:
1397:
1394:
1392:
1389:
1387:
1384:
1383:
1381:
1379:
1375:
1369:
1366:
1364:
1361:
1357:
1354:
1352:
1349:
1348:
1347:
1346:Point process
1344:
1342:
1339:
1337:
1334:
1332:
1329:
1325:
1322:
1320:
1317:
1316:
1315:
1312:
1310:
1307:
1305:
1304:Gibbs measure
1302:
1300:
1297:
1295:
1292:
1291:
1289:
1285:
1279:
1276:
1274:
1271:
1269:
1266:
1264:
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1259:
1256:
1252:
1249:
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1244:
1242:
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1237:
1234:
1233:
1232:
1229:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1208:
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1196:
1193:
1191:
1188:
1186:
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1181:
1178:
1176:
1173:
1171:
1168:
1166:
1163:
1161:
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1156:
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1149:
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1144:
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1140:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1093:Itô diffusion
1091:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1069:
1068:Gamma process
1066:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
999:
997:
994:
993:
992:
989:
985:
982:
981:
980:
977:
975:
972:
970:
967:
966:
964:
962:
958:
950:
947:
945:
942:
940:
939:Self-avoiding
937:
935:
932:
931:
930:
927:
925:
924:Moran process
922:
920:
917:
915:
912:
910:
907:
905:
902:
900:
897:
895:
892:
891:
889:
887:
886:Discrete time
883:
879:
872:
867:
865:
860:
858:
853:
852:
849:
837:
834:
829:
823:
815:
809:
805:
804:
796:
793:
788:
782:
778:
773:
772:
763:
760:
755:
749:
742:
741:
733:
730:
723:
721:
715:
699:
690:
687:
684:
681:
673:
667:
664:
656:
647:
644:
641:
633:
627:
624:
616:
608:
604:
600:
595:
591:
587:
582:
577:
573:
569:
563:
557:
549:
532:
517:
509:
505:
501:
496:
492:
486:
481:
477:
473:
467:
461:
458:
450:
433:
429:
425:
411:
401:
393:
390:
386:
380:
376:
372:
366:
358:
341:
337:
336:
335:
319:
316:
312:
304:
303:time constant
286:
282:
278:
269:
261:
257:
239:
233:
225:
223:
217:
213:
209:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
158:
154:
150:
146:
142:
138:
134:
130:
126:
123:
119:
115:
111:
107:
103:
99:
95:
91:
87:
86:
85:
83:
79:
71:
69:
67:
62:
60:
56:
52:
48:
44:
43:Andrey Markov
40:
37:(named after
36:
30:
19:
2031:Econometrics
1993:Wiener space
1881:Itô integral
1782:Inequalities
1671:Self-similar
1641:Gauss–Markov
1640:
1631:Exchangeable
1611:Càdlàg paths
1547:Risk process
1499:LIBOR market
1368:Random graph
1363:Random field
1175:Superprocess
1113:Lévy process
1108:Jump process
1083:Hunt process
919:Markov chain
836:
802:
795:
770:
762:
739:
732:
719:
338:Exponential
235:
221:
211:
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199:
195:
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93:
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2076:Ruin theory
2014:Disciplines
1886:Itô's lemma
1661:Predictable
1336:Percolation
1319:Potts model
1314:Ising model
1278:White noise
1236:Differences
1098:Itô process
1038:Cox process
934:Loop-erased
929:Random walk
2086:Statistics
1866:Filtration
1767:Kolmogorov
1751:Blumenthal
1676:Stationary
1616:Continuous
1604:Properties
1489:Hull–White
1231:Martingale
1118:Local time
1006:Fractional
984:pure birth
753:026218253X
724:References
206:)), where
1998:Classical
1011:Geometric
1001:Excursion
822:cite book
691:β
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674:σ
668:β
657:⋅
648:β
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628:β
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2119:Category
2003:Abstract
1537:Bühlmann
1143:Compound
426:A power
238:variance
1626:Ergodic
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1356:Poisson
1016:Meander
139:, then
100:, then
1966:Tanaka
1651:Mixing
1646:Markov
1519:Wilkie
1484:Ho–Lee
1479:Heston
1251:Super-
996:Bridge
944:Biased
810:
783:
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1819:Tools
1595:M/M/c
1590:M/M/1
1585:M/G/1
1575:Fluid
1241:Local
779:–95.
744:(PDF)
1771:Lévy
1570:Bulk
1454:Chen
1246:Sub-
1204:Both
828:link
808:ISBN
781:ISBN
748:ISBN
301:and
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