426:, located in a 2D grid at (0,0), (0,2), (2,2), and (2,0), respectively. Suppose each random variable can take on the value of -1 or 1, and the probability of each random variable's value depends on its immediately adjacent neighbours. This is a simple example of a discrete random field.
946:
1048:(RVE) of deterministic continuum physics. The second type of random field that appears in continuum theories are those of dependent quantities (temperature, displacement, velocity, deformation, rotation, body and surface forces, stress, etc.).
682:
963:, values in a random field are often spatially correlated. For example, adjacent values (i.e. values with adjacent indices) do not differ as much as values that are further apart. This is an example of a
780:
774:. In other words, the probability that a random variable assumes a value depends on its immediate neighboring random variables. The probability of a random variable in an MRF is given by
1439:
456:
can take on might be defined over a continuous domain. In larger grids, it can also be useful to think of the random field as a "function valued" random variable as described above. In
196:
119:
1974:
267:
55:
752:
725:
512:
1798:
1044:(SVE), which is a spatial box over which properties can be averaged; when the SVE becomes sufficiently large, its properties become deterministic and one recovers the
454:
424:
397:
370:
343:
297:
84:
2401:
1394:
1931:
1911:
1040:
in which the random fields correspond to naturally spatially varying properties. This leads to tensor-valued random fields in which the key role is played by a
772:
2315:
311:
In its discrete version, a random field is a list of random numbers whose indices are identified with a discrete set of points in a space (for example, n-
2232:
2242:
1916:
1301:
1926:
2284:
1999:
2181:
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1984:
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998:
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2019:
1989:
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2191:
1411:
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where the sum (can be an integral) is over the possible values of k. It is sometimes difficult to compute this quantity exactly.
974:
A common use of random fields is in the generation of computer graphics, particularly those that mimic natural surfaces such as
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2356:
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1959:
1756:
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1954:
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2141:
1553:
1469:
1380:
1045:
2431:
2067:
1134:"A two-dimensional approach to quantify stratigraphic uncertainty from borehole data using non-homogeneous random fields"
2426:
1087:
1002:
994:
2029:
941:{\displaystyle P(X_{i}=x_{i}|\partial _{i})={\frac {P(X_{i}=x_{i},\partial _{i})}{\sum _{k}P(X_{i}=k,\partial _{i})}},}
1613:
1558:
1474:
1082:
2361:
2351:
1994:
1964:
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where sometimes nearest neighbor interactions are only included as a simplification to better understand the model.
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125:
with some restriction on its index set. That is, by modern definitions, a random field is a generalization of a
1668:
1643:
1248:
484:
461:
2386:
1969:
1504:
1210:
Vitells, Ofer; Gross, Eilam (2011). "Estimating the significance of a signal in a multi-dimensional search".
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130:
2014:
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31:
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1936:
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457:
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1946:
1856:
1746:
730:
677:{\displaystyle P(X_{i}=x_{i}|X_{j}=x_{j},i\neq j)=P(X_{i}=x_{i}|X_{j}=x_{j},j\in \partial _{i}),\,}
476:
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structure, many different types of which may be modeled in a random field. One example is the
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2009:
2626:
2513:
2396:
1766:
1741:
1690:
1618:
1541:
1494:
1310:
1299:
Besag, J. E. (1974). "Spatial
Interaction and the Statistical Analysis of Lattice Systems".
1229:
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28:
is a random function over an arbitrary domain (usually a multi-dimensional space such as
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1771:
1593:
1509:
1449:
1009:
significant activation. More generally, random fields can be used to correct for the
495:
proposed an approximation method relying on the relation between MRFs and Gibbs RFs.
1163:"A Three-Dimensional Statistical Analysis for CBF Activation Studies in Human Brain"
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2004:
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21:
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1979:
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1067:
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121:(or some other domain). It is also sometimes thought of as a synonym for a
1196:
142:
1932:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
1372:
1161:
Worsley, K. J.; Evans, A. C.; Marrett, S.; Neelin, P. (November 1992).
1062:
982:. Random fields have been also used in subsurface ground models as in
134:
17:
1001:, statistical analysis of random fields are one common alternative to
1036:
Random fields are of great use in studying natural processes by the
137:
valued "time" but can instead take values that are multidimensional
1224:
1072:
1356:
Multiparameter
Processes : An Introduction to Random Fields
1376:
174:
1912:
Autoregressive conditional heteroskedasticity (ARCH) model
1440:
Independent and identically distributed random variables
1917:
Autoregressive integrated moving average (ARIMA) model
783:
760:
733:
693:
515:
475:
Several kinds of random fields exist, among them the
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324:
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63:
34:
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418:
391:
364:
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291:
261:
190:
113:
78:
49:
1253:Tensor-Valued Random Fields for Continuum Physics
129:where the underlying parameter need no longer be
1799:Stochastic chains with memory of variable length
1013:in statistical testing, where the domain is the
1167:Journal of Cerebral Blood Flow & Metabolism
1388:
1322:Griffeath, David (1976). "Random Fields". In
8:
1280:Adler, R. J. & Taylor, Jonathan (2007).
318:). Suppose there are four random variables,
256:
231:
86:that takes on a random value at each point
1927:Autoregressive–moving-average (ARMA) model
1395:
1381:
1373:
191:{\displaystyle (\Omega ,{\mathcal {F}},P)}
1223:
1178:
1145:
923:
904:
888:
873:
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847:
834:
822:
813:
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782:
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732:
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673:
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567:
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464:, one that takes on random values over a
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383:
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356:
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329:
323:
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238:
229:
173:
172:
161:
105:
101:
100:
91:
62:
41:
37:
36:
33:
1302:Journal of the Royal Statistical Society
202:-valued random field is a collection of
1111:. World Scientific Publishing Company.
1099:
2233:Doob's martingale convergence theorems
460:the notion is generalized to a random
1985:Constant elasticity of variance (CEV)
1975:Chan–Karolyi–Longstaff–Sanders (CKLS)
1109:Random Fields: Analysis and Synthesis
991:task-related functional brain imaging
114:{\displaystyle x\in \mathbb {R} ^{n}}
7:
1234:10.1016/j.astropartphys.2011.08.005
1003:correction for multiple comparisons
2472:Skorokhod's representation theorem
2253:Law of large numbers (weak/strong)
1315:10.1111/j.2517-6161.1974.tb00999.x
920:
870:
819:
735:
658:
166:
14:
2442:Martingale representation theorem
2487:Stochastic differential equation
2377:Doob's optional stopping theorem
2372:Doob–Meyer decomposition theorem
429:More generally, the values each
262:{\displaystyle \{F_{t}:t\in T\}}
50:{\displaystyle \mathbb {R} ^{n}}
2357:Convergence of random variables
2243:Fisher–Tippett–Gnedenko theorem
1955:Binomial options pricing model
1255:. Cambridge University Press.
929:
897:
879:
840:
828:
814:
787:
708:
694:
667:
621:
594:
585:
546:
519:
185:
163:
73:
67:
1:
2422:Kolmogorov continuity theorem
2258:Law of the iterated logarithm
1046:representative volume element
747:{\displaystyle \partial _{i}}
57:). That is, it is a function
2427:Kolmogorov extension theorem
2106:Generalized queueing network
1614:Interacting particle systems
1330:; Knapp, Anthony W. (eds.).
1147:10.1016/j.enggeo.2023.107001
1088:Stochastic cellular automata
754:is the set of neighbors of
1559:Continuous-time random walk
1083:Interacting particle system
1032:Tensor-valued random fields
720:{\displaystyle (x_{j})_{j}}
2682:
2567:Extreme value theory (EVT)
2367:Doob decomposition theorem
1659:Ornstein–Uhlenbeck process
1430:Chinese restaurant process
1336:(2nd ed.). Springer.
1282:Random Fields and Geometry
1042:statistical volume element
687:for each choice of values
217:. That is, a random field
2635:
2447:Optional stopping theorem
2248:Large deviation principle
2000:Heath–Jarrow–Morton (HJM)
1937:Moving-average (MA) model
1922:Autoregressive (AR) model
1747:Hidden Markov model (HMM)
1681:Schramm–Loewner evolution
1333:Denumerable Markov Chains
1249:Ostoja-Starzewski, Martin
303:-valued random variable.
210:indexed by elements in a
2362:Doléans-Dade exponential
2192:Progressively measurable
1990:Cox–Ingersoll–Ross (CIR)
1107:Vanmarcke, Erik (2010).
485:conditional random field
2582:Mathematical statistics
2572:Large deviations theory
2402:Infinitesimal generator
2263:Maximal ergodic theorem
2182:Piecewise-deterministic
1784:Random dynamical system
1649:Markov additive process
2417:Karhunen–Loève theorem
2352:Cameron–Martin formula
2316:Burkholder–Davis–Gundy
1711:Variance gamma process
1247:Malyarenko, Anatoliy;
1180:10.1038/jcbfm.1992.127
1020:They are also used in
942:
768:
748:
721:
678:
450:
420:
393:
366:
339:
293:
263:
192:
115:
80:
51:
2547:Actuarial mathematics
2509:Uniform integrability
2504:Stratonovich integral
2432:Lévy–Prokhorov metric
2336:Marcinkiewicz–Zygmund
2223:Central limit theorem
1825:Gaussian random field
1654:McKean–Vlasov process
1574:Dyson Brownian motion
1435:Galton–Watson process
1212:Astroparticle Physics
1132:Cardenas, IC (2023).
1011:look-elsewhere effect
1005:to find regions with
943:
769:
749:
722:
679:
489:Gaussian random field
451:
449:{\displaystyle X_{i}}
421:
419:{\displaystyle X_{4}}
394:
392:{\displaystyle X_{3}}
367:
365:{\displaystyle X_{2}}
340:
338:{\displaystyle X_{1}}
294:
292:{\displaystyle F_{t}}
264:
193:
116:
81:
52:
2622:Time series analysis
2577:Mathematical finance
2462:Reflection principle
1789:Regenerative process
1589:Fleming–Viot process
1404:Stochastic processes
781:
758:
731:
691:
513:
503:An MRF exhibits the
458:quantum field theory
433:
403:
376:
349:
322:
276:
228:
160:
90:
79:{\displaystyle f(x)}
61:
32:
2617:Stochastic analysis
2457:Quadratic variation
2452:Prokhorov's theorem
2387:Feynman–Kac formula
1857:Markov random field
1505:Birth–death process
1138:Engineering Geology
477:Markov random field
2587:Probability theory
2467:Skorokhod integral
2437:Malliavin calculus
2020:Korn-Kreer-Lenssen
1904:Time series models
1867:Pitman–Yor process
1352:Davar Khoshnevisan
1078:Stochastic process
1038:Monte Carlo method
1024:applications (see
989:, particularly in
938:
893:
764:
744:
717:
674:
499:Example properties
481:Gibbs random field
466:space of functions
446:
416:
389:
362:
335:
289:
259:
188:
141:or points on some
127:stochastic process
123:stochastic process
111:
76:
47:
2666:Spatial processes
2653:
2652:
2607:Signal processing
2326:Doob's upcrossing
2321:Doob's martingale
2285:Engelbert–Schmidt
2228:Donsker's theorem
2162:Feller-continuous
2030:Rendleman–Bartter
1820:Dirichlet process
1737:Branching process
1706:Telegraph process
1599:Geometric process
1579:Empirical process
1569:Diffusion process
1425:Branching process
1420:Bernoulli process
1291:978-0-387-48112-8
959:When used in the
933:
884:
767:{\displaystyle i}
212:topological space
155:probability space
149:Formal definition
2673:
2627:Machine learning
2514:Usual hypotheses
2397:Girsanov theorem
2382:Dynkin's formula
2147:Continuous paths
2055:Actuarial models
1995:Garman–Kohlhagen
1965:Black–Karasinski
1960:Black–Derman–Toy
1947:Financial models
1813:Fields and other
1742:Gaussian process
1691:Sigma-martingale
1495:Additive process
1397:
1390:
1383:
1374:
1369:
1347:
1318:
1295:
1267:
1266:
1244:
1238:
1237:
1227:
1207:
1201:
1200:
1182:
1158:
1152:
1151:
1149:
1129:
1123:
1122:
1104:
1026:graphical models
1022:machine learning
1017:being searched.
961:natural sciences
947:
945:
944:
939:
934:
932:
928:
927:
909:
908:
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811:
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619:
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559:
558:
549:
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543:
531:
530:
470:Feynman integral
455:
453:
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425:
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298:
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268:
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243:
242:
221:is a collection
208:random variables
197:
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178:
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120:
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112:
110:
109:
104:
85:
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77:
56:
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53:
48:
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45:
40:
2681:
2680:
2676:
2675:
2674:
2672:
2671:
2670:
2656:
2655:
2654:
2649:
2631:
2592:Queueing theory
2535:
2477:Skorokhod space
2340:
2331:Kunita–Watanabe
2302:
2268:Sanov's theorem
2238:Ergodic theorem
2211:
2207:Time-reversible
2125:
2088:Queueing models
2082:
2078:Sparre–Anderson
2068:Cramér–Lundberg
2049:
2035:SABR volatility
1941:
1898:
1850:Boolean network
1808:
1794:Renewal process
1725:
1674:Non-homogeneous
1664:Poisson process
1554:Contact process
1517:Brownian motion
1487:Continuous time
1481:
1475:Maximal entropy
1406:
1401:
1366:
1350:
1344:
1324:Kemeny, John G.
1321:
1298:
1292:
1279:
1276:
1274:Further reading
1271:
1270:
1263:
1246:
1245:
1241:
1209:
1208:
1204:
1160:
1159:
1155:
1131:
1130:
1126:
1119:
1106:
1105:
1101:
1096:
1054:
1034:
1015:parameter space
957:
919:
900:
883:
869:
856:
843:
836:
818:
803:
790:
779:
778:
756:
755:
734:
729:
728:
707:
697:
689:
688:
657:
638:
625:
610:
597:
563:
550:
535:
522:
511:
510:
505:Markov property
501:
436:
431:
430:
406:
401:
400:
379:
374:
373:
352:
347:
346:
325:
320:
319:
316:Euclidean space
309:
279:
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273:
234:
226:
225:
158:
157:
151:
99:
88:
87:
59:
58:
35:
30:
29:
12:
11:
5:
2679:
2677:
2669:
2668:
2658:
2657:
2651:
2650:
2648:
2647:
2642:
2640:List of topics
2636:
2633:
2632:
2630:
2629:
2624:
2619:
2614:
2609:
2604:
2599:
2597:Renewal theory
2594:
2589:
2584:
2579:
2574:
2569:
2564:
2562:Ergodic theory
2559:
2554:
2552:Control theory
2549:
2543:
2541:
2537:
2536:
2534:
2533:
2532:
2531:
2526:
2516:
2511:
2506:
2501:
2496:
2495:
2494:
2484:
2482:Snell envelope
2479:
2474:
2469:
2464:
2459:
2454:
2449:
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2439:
2434:
2429:
2424:
2419:
2414:
2409:
2404:
2399:
2394:
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2379:
2374:
2369:
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2354:
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2341:
2339:
2338:
2333:
2328:
2323:
2318:
2312:
2310:
2304:
2303:
2301:
2300:
2281:Borel–Cantelli
2270:
2265:
2260:
2255:
2250:
2245:
2240:
2235:
2230:
2225:
2219:
2217:
2216:Limit theorems
2213:
2212:
2210:
2209:
2204:
2199:
2194:
2189:
2184:
2179:
2174:
2169:
2164:
2159:
2154:
2149:
2144:
2139:
2133:
2131:
2127:
2126:
2124:
2123:
2118:
2113:
2108:
2103:
2098:
2092:
2090:
2084:
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2081:
2080:
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2070:
2065:
2059:
2057:
2051:
2050:
2048:
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2042:
2037:
2032:
2027:
2022:
2017:
2012:
2007:
2002:
1997:
1992:
1987:
1982:
1977:
1972:
1967:
1962:
1957:
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1939:
1934:
1929:
1924:
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1899:
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1891:
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1885:
1884:
1879:
1869:
1864:
1859:
1854:
1853:
1852:
1847:
1837:
1835:Hopfield model
1832:
1827:
1822:
1816:
1814:
1810:
1809:
1807:
1806:
1801:
1796:
1791:
1786:
1781:
1780:
1779:
1774:
1769:
1764:
1754:
1752:Markov process
1749:
1744:
1739:
1733:
1731:
1727:
1726:
1724:
1723:
1721:Wiener sausage
1718:
1716:Wiener process
1713:
1708:
1703:
1698:
1696:Stable process
1693:
1688:
1686:Semimartingale
1683:
1678:
1677:
1676:
1671:
1661:
1656:
1651:
1646:
1641:
1636:
1631:
1629:Jump diffusion
1626:
1621:
1616:
1611:
1606:
1604:Hawkes process
1601:
1596:
1591:
1586:
1584:Feller process
1581:
1576:
1571:
1566:
1561:
1556:
1551:
1549:Cauchy process
1546:
1545:
1544:
1539:
1534:
1529:
1524:
1514:
1513:
1512:
1502:
1500:Bessel process
1497:
1491:
1489:
1483:
1482:
1480:
1479:
1478:
1477:
1472:
1467:
1462:
1452:
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1442:
1437:
1432:
1427:
1422:
1416:
1414:
1408:
1407:
1402:
1400:
1399:
1392:
1385:
1377:
1371:
1370:
1364:
1348:
1342:
1319:
1309:(2): 192–236.
1296:
1290:
1275:
1272:
1269:
1268:
1261:
1239:
1202:
1173:(6): 900–918.
1153:
1124:
1118:978-9812563538
1117:
1098:
1097:
1095:
1092:
1091:
1090:
1085:
1080:
1075:
1070:
1065:
1060:
1053:
1050:
1033:
1030:
993:studies using
956:
953:
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931:
926:
922:
918:
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903:
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846:
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839:
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821:
816:
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806:
802:
797:
793:
789:
786:
763:
741:
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714:
710:
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700:
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685:
684:
672:
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664:
660:
656:
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637:
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623:
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584:
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443:
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386:
382:
359:
355:
332:
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308:
305:
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270:
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258:
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187:
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176:
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108:
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98:
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72:
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44:
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13:
10:
9:
6:
4:
3:
2:
2678:
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2527:
2525:
2522:
2521:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2499:Stopping time
2497:
2493:
2490:
2489:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
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2398:
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2334:
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2329:
2327:
2324:
2322:
2319:
2317:
2314:
2313:
2311:
2309:
2305:
2298:
2294:
2290:
2289:Hewitt–Savage
2286:
2282:
2278:
2274:
2273:Zero–one laws
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
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2200:
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2132:
2128:
2122:
2119:
2117:
2114:
2112:
2109:
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2102:
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2097:
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2093:
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2089:
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2079:
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2033:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1970:Black–Scholes
1968:
1966:
1963:
1961:
1958:
1956:
1953:
1952:
1950:
1948:
1944:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
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1907:
1905:
1901:
1895:
1892:
1890:
1887:
1883:
1880:
1878:
1875:
1874:
1873:
1872:Point process
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1851:
1848:
1846:
1843:
1842:
1841:
1838:
1836:
1833:
1831:
1830:Gibbs measure
1828:
1826:
1823:
1821:
1818:
1817:
1815:
1811:
1805:
1802:
1800:
1797:
1795:
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1655:
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1650:
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1645:
1642:
1640:
1637:
1635:
1632:
1630:
1627:
1625:
1622:
1620:
1619:ItĂ´ diffusion
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1595:
1594:Gamma process
1592:
1590:
1587:
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1582:
1580:
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1567:
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1507:
1506:
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1501:
1498:
1496:
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1488:
1484:
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1473:
1471:
1468:
1466:
1465:Self-avoiding
1463:
1461:
1458:
1457:
1456:
1453:
1451:
1450:Moran process
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1417:
1415:
1413:
1412:Discrete time
1409:
1405:
1398:
1393:
1391:
1386:
1384:
1379:
1378:
1375:
1367:
1365:0-387-95459-7
1361:
1357:
1353:
1349:
1345:
1343:0-387-90177-9
1339:
1335:
1334:
1329:
1328:Snell, Laurie
1325:
1320:
1316:
1312:
1308:
1304:
1303:
1297:
1293:
1287:
1283:
1278:
1277:
1273:
1264:
1262:9781108429856
1258:
1254:
1250:
1243:
1240:
1235:
1231:
1226:
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1217:
1213:
1206:
1203:
1198:
1194:
1190:
1186:
1181:
1176:
1172:
1168:
1164:
1157:
1154:
1148:
1143:
1139:
1135:
1128:
1125:
1120:
1114:
1110:
1103:
1100:
1093:
1089:
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1069:
1066:
1064:
1061:
1059:
1056:
1055:
1051:
1049:
1047:
1043:
1039:
1031:
1029:
1027:
1023:
1018:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
983:
981:
977:
972:
970:
966:
962:
954:
952:
935:
924:
916:
913:
910:
905:
901:
894:
889:
885:
874:
866:
861:
857:
853:
848:
844:
837:
831:
823:
808:
804:
800:
795:
791:
784:
777:
776:
775:
761:
739:
712:
702:
698:
670:
662:
654:
651:
648:
643:
639:
635:
630:
626:
615:
611:
607:
602:
598:
591:
588:
582:
579:
576:
573:
568:
564:
560:
555:
551:
540:
536:
532:
527:
523:
516:
509:
508:
507:
506:
498:
496:
494:
490:
486:
482:
478:
473:
471:
467:
463:
459:
441:
437:
427:
411:
407:
384:
380:
357:
353:
330:
326:
317:
314:
306:
304:
302:
284:
280:
253:
250:
247:
244:
239:
235:
224:
223:
222:
220:
216:
213:
209:
205:
201:
182:
179:
169:
156:
148:
146:
144:
140:
136:
132:
128:
124:
106:
96:
93:
70:
64:
42:
27:
23:
19:
2557:Econometrics
2519:Wiener space
2407:ItĂ´ integral
2308:Inequalities
2197:Self-similar
2167:Gauss–Markov
2157:Exchangeable
2137:CĂ dlĂ g paths
2073:Risk process
2025:LIBOR market
1894:Random graph
1889:Random field
1888:
1701:Superprocess
1639:LĂ©vy process
1634:Jump process
1609:Hunt process
1445:Markov chain
1358:. Springer.
1355:
1332:
1306:
1305:. Series B.
1300:
1284:. Springer.
1281:
1252:
1242:
1215:
1211:
1205:
1170:
1166:
1156:
1137:
1127:
1108:
1102:
1041:
1035:
1019:
1006:
987:neuroscience
984:
973:
958:
955:Applications
950:
727:. Here each
686:
502:
493:Julian Besag
474:
428:
310:
300:
271:
218:
214:
203:
199:
152:
26:random field
25:
15:
2602:Ruin theory
2540:Disciplines
2412:ItĂ´'s lemma
2187:Predictable
1862:Percolation
1845:Potts model
1840:Ising model
1804:White noise
1762:Differences
1624:ItĂ´ process
1564:Cox process
1460:Loop-erased
1455:Random walk
1218:: 230–234.
969:Ising model
491:. In 1974,
487:(CRF), and
313:dimensional
272:where each
22:mathematics
2612:Statistics
2392:Filtration
2293:Kolmogorov
2277:Blumenthal
2202:Stationary
2142:Continuous
2130:Properties
2015:Hull–White
1757:Martingale
1644:Local time
1532:Fractional
1510:pure birth
1094:References
1058:Covariance
965:covariance
462:functional
2524:Classical
1537:Geometric
1527:Excursion
1225:1105.4355
1189:0271-678X
1068:Variogram
921:∂
886:∑
871:∂
820:∂
736:∂
659:∂
655:∈
580:≠
251:∈
167:Ω
97:∈
2660:Category
2645:Category
2529:Abstract
2063:BĂĽhlmann
1669:Compound
1354:(2002).
1251:(2019).
1052:See also
307:Examples
206:-valued
153:Given a
143:manifold
2152:Ergodic
2040:VašĂÄŤek
1882:Poisson
1542:Meander
1197:1400644
1063:Kriging
479:(MRF),
139:vectors
135:integer
18:physics
2492:Tanaka
2177:Mixing
2172:Markov
2045:Wilkie
2010:Ho–Lee
2005:Heston
1777:Super-
1522:Bridge
1470:Biased
1362:
1340:
1288:
1259:
1195:
1187:
1115:
399:, and
299:is an
2345:Tools
2121:M/M/c
2116:M/M/1
2111:M/G/1
2101:Fluid
1767:Local
1220:arXiv
1073:Resel
1007:truly
980:earth
976:water
468:(see
198:, an
2297:LĂ©vy
2096:Bulk
1980:Chen
1772:Sub-
1730:Both
1360:ISBN
1338:ISBN
1286:ISBN
1257:ISBN
1193:PMID
1185:ISSN
1113:ISBN
999:fMRI
978:and
472:).
131:real
24:, a
20:and
1877:Cox
1311:doi
1230:doi
1175:doi
1142:doi
1028:).
997:or
995:PET
985:In
133:or
16:In
2662::
2295:,
2291:,
2287:,
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.