2845:
2855:
412:
539:
552:
in the classical version of the central limit theorem. More generally, if the u.a.n. condition is satisfied via a scaling of identically distributed random variables (with not necessarily finite second moment), then the weak convergence is to a
140:
to find families of probability distributions that might be natural choices for certain models or applications. Infinitely divisible distributions play an important role in probability theory in the context of limit theorems.
296:
431:
621:
40:
2889:
1283:
1237:
1191:
1143:
1059:
1503:
544:
Thus, for example, if the uniform asymptotic negligibility (u.a.n.) condition is satisfied via an appropriate scaling of identically distributed random variables with finite
1097:
1632:
2879:
48:
2884:
2115:
2023:
2810:
407:{\displaystyle {\begin{array}{cccc}X_{11}\\X_{21}&X_{22}\\X_{31}&X_{32}&X_{33}\\\vdots &\vdots &\vdots &\ddots \end{array}}}
2676:
1888:
1647:
1496:
742:
of the corresponding difference on any interval not overlapping with , and similarly for any finite number of mutually non-overlapping intervals.
2571:
2335:
1294:
2009:
1384:
2330:
2274:
1934:
1572:
534:{\displaystyle \lim _{n\to \infty }\,\max _{1\leq k\leq n}\;P(\left|X_{nk}\right|>\varepsilon )=0{\text{ for every }}\varepsilon >0.}
2080:
2616:
2350:
2203:
1878:
1622:
2858:
2075:
2848:
2520:
2496:
1489:
2717:
2345:
2594:
2555:
2527:
2501:
2419:
1768:
1516:
1434:
1409:
208:
571:
2705:
2671:
2537:
2532:
2377:
2185:
1883:
1637:
2455:
2368:
2340:
2249:
2198:
2172:
2070:
1853:
1818:
1193:
satisfies a number of conditions of continuity and monotonicity. Moreover, if a family of infinitely divisible distributions
2469:
2386:
2223:
1970:
1848:
1823:
1687:
1682:
1677:
193:
2147:
1657:
1652:
2785:
2651:
2359:
2208:
2140:
2125:
2018:
1992:
1924:
1763:
1594:
1579:
1304:
1066:
239:
2302:
2681:
2621:
2611:
2228:
1929:
1788:
1453:
Domínguez-Molina, J.A.; Rocha-Arteaga, A. (2007) "On the
Infinite Divisibility of some Skewed Symmetric Distributions".
2030:
1773:
1702:
2666:
2661:
2606:
2542:
2307:
2085:
1982:
1567:
1299:
418:
2486:
2294:
182:
2800:
2576:
2395:
2177:
2130:
1999:
1975:
1955:
1798:
1672:
1552:
2805:
1748:
2589:
2550:
2424:
2261:
2105:
2050:
1948:
1912:
1783:
129:
2491:
2279:
2045:
2004:
1919:
1873:
1813:
1778:
1667:
1562:
1512:
739:
687:
287:
32:
1604:
17:
2790:
2732:
2403:
2190:
2100:
2055:
2040:
1960:
1858:
1808:
1803:
1584:
1242:
1196:
1150:
1102:
178:
170:
1239:
satisfies these continuity and monotonicity conditions, there exists (uniquely in law) an additive process
2656:
2644:
2633:
2515:
2411:
2218:
1662:
1642:
1547:
1070:
671:
562:
232:
228:
201:
197:
1018:
2780:
2737:
2581:
2256:
2110:
2090:
1987:
1557:
668:
631:
251:
220:
212:
2830:
2825:
2820:
2815:
2752:
2722:
2601:
2244:
2135:
2035:
1738:
1697:
1692:
1589:
224:
189:
158:
2764:
2289:
2269:
2239:
2213:
2167:
2095:
1907:
1843:
554:
549:
162:
154:
150:
2795:
2284:
2065:
2060:
1965:
1902:
1897:
1753:
1743:
1627:
653:
166:
28:
301:
2693:
2120:
1863:
1793:
1758:
1707:
1430:
1405:
1380:
1357:
987:
174:
1076:
1868:
1542:
1458:
1349:
1013:
1007:
558:
125:
124:
The concept of infinite divisibility of probability distributions was introduced in 1929 by
39:
if it can be expressed as the probability distribution of the sum of an arbitrary number of
929:
44:
1467:
Steutel, F. W. (1979), "Infinite
Divisibility in Theory and Practice" (with discussion),
649:
643:
235:
of a random variable having a
Student's t-distribution is also not infinitely divisible.
290:
uniformly asymptotically negligible (u.a.n.) random variables within a triangular array
1941:
648:
Every infinitely divisible probability distribution corresponds in a natural way to a
2873:
2564:
2312:
1599:
1481:
1353:
1340:
Thorin, Olof (1977). "On the infinite divisibility of the lognormal distribution".
909:
is an infinitely divisible distribution, we can construct a Lévy process {
133:
1462:
137:
1361:
250:
Infinitely divisible distributions appear in a broad generalization of the
204:
whose only possible outcome is 0 is also (trivially) infinitely divisible.
149:
Examples of continuous distributions that are infinitely divisible are the
626:
the weak convergence of the sum is to the
Poisson distribution with mean
545:
242:
is infinitely divisible; this follows immediately from the definition.
1404:(2nd ed.). Wiley. volume 2, chapter 28, page 368.
1062:
1476:
Infinite
Divisibility of Probability Distributions on the Real Line
219:
infinitely divisible, nor are any other distributions with bounded
1485:
616:{\displaystyle \lim _{n\rightarrow \infty }np_{n}=\lambda ,}
51:
of any infinitely divisible distribution is then called an
998: > 0 are handled via a continuity argument.
920: ≥ 0 } from it. For any interval where
62:
is infinitely divisible if, for every positive integer
756: ≥ 0 } is a Lévy process then, for any
1427:
Lévy processes and infinitely divisible distributions
1377:
Lévy
Processes and Infinitely Divisible Distributions
1245:
1199:
1153:
1145:
be its family of infinitely divisible distributions.
1105:
1079:
1021:
574:
434:
299:
2773:
2731:
2632:
2468:
2446:
2437:
2321:
2156:
1832:
1729:
1720:
1613:
1533:
1524:
1073:) has an infinitely divisible distribution for any
188:Among the discrete distributions, examples are the
1400:Johnson, N.L.; Kotz, S.; Balakrishnan, N. (1995).
1277:
1231:
1185:
1137:
1091:
1053:
615:
533:
421:— an infinitely divisible distribution. The
406:
1379:. Cambridge University Press. pp. 31, 148.
576:
565:where the u.a.n. condition is satisfied through
453:
436:
2890:Infinitely divisible probability distributions
1429:. Cambridge University Press. pp. 31–68.
58:More rigorously, the probability distribution
1497:
18:Infinitely divisible probability distribution
8:
1260:
1246:
1214:
1200:
1168:
1154:
1120:
1106:
1036:
1022:
423:uniformly asymptotically negligible (u.a.n.)
53:infinitely divisible characteristic function
2443:
1726:
1530:
1504:
1490:
1482:
474:
1263:
1253:
1244:
1217:
1207:
1198:
1171:
1161:
1152:
1123:
1113:
1104:
1078:
1039:
1029:
1020:
598:
579:
573:
517:
489:
456:
451:
439:
433:
372:
360:
348:
334:
322:
308:
300:
298:
231:mentioned above. The distribution of the
1474:Steutel, F. W. and Van Harn, K. (2003),
1322:
1320:
1316:
41:independent and identically distributed
1278:{\displaystyle \{\mu _{t}\}_{t\geq 0}}
1232:{\displaystyle \{\mu _{t}\}_{t\geq 0}}
1186:{\displaystyle \{\mu _{t}\}_{t\geq 0}}
1138:{\displaystyle \{\mu _{t}\}_{t\geq 0}}
769:will be infinitely divisible: for any
630:as shown by the familiar proof of the
7:
2854:
760: ≥ 0, the random variable
2880:Theory of probability distributions
1469:Scandinavian Journal of Statistics.
1402:Continuous Univariate Distributions
1054:{\displaystyle \{X_{t}\}_{t\geq 0}}
2885:Types of probability distributions
1455:Statistics and Probability Letters
586:
446:
25:
957:to have the same distribution as
548:, the weak convergence is to the
2853:
2844:
2843:
894:is infinitely divisible for any
161:, and all other members of the
130:decomposition of a distribution
1354:10.1080/03461238.1977.10405635
1342:Scandinavian Actuarial Journal
1330:, Griffin, London. p. 107
583:
508:
478:
443:
194:negative binomial distribution
1:
1305:Compound Poisson distribution
240:compound Poisson distribution
258:→ +∞ of the sum
1300:Indecomposable distribution
928: > 0 equals a
720:means that that difference
667: ≥ 0 } with
557:. On the other hand, for a
2906:
2677:Wrapped asymmetric Laplace
1648:Extended negative binomial
1005:
641:
563:Bernoulli random variables
561:of independent (unscaled)
417:approaches — in the
117:has the same distribution
2839:
2336:Generalized extreme value
2116:Relativistic Breit–Wigner
1513:Probability distributions
1463:10.1016/j.spl.2006.09.014
1067:continuous in probability
1457:, 77 (6), 644–648
1328:Characteristic Functions
1285:with this distribution.
1069:stochastic process with
688:probability distribution
183:Student's t-distribution
70:i.i.d. random variables
33:probability distribution
2331:Generalized chi-squared
2275:Normal-inverse Gaussian
1092:{\displaystyle t\geq 0}
652:. A Lévy process is a
179:Log-normal distribution
171:chi-square distribution
165:family, as well as the
49:characteristic function
2643:Univariate (circular)
2204:Generalized hyperbolic
1633:Conway–Maxwell–Poisson
1623:Beta negative binomial
1425:Sato, Ken-Ito (1999).
1375:Sato, Ken-iti (1999).
1279:
1233:
1187:
1139:
1093:
1071:independent increments
1055:
905:On the other hand, if
718:independent increments
672:independent increments
617:
535:
425:condition is given by
408:
229:one-point distribution
202:one-point distribution
198:geometric distribution
2688:Bivariate (spherical)
2186:Kaniadakis κ-Gaussian
1280:
1234:
1188:
1140:
1094:
1056:
618:
536:
519: for every
409:
252:central limit theorem
213:binomial distribution
2753:Dirac delta function
2700:Bivariate (toroidal)
2657:Univariate von Mises
2528:Multivariate Laplace
2420:Shifted log-logistic
1769:Continuous Bernoulli
1243:
1197:
1151:
1103:
1077:
1019:
632:law of small numbers
572:
432:
297:
209:uniform distribution
190:Poisson distribution
37:infinitely divisible
2801:Natural exponential
2706:Bivariate von Mises
2672:Wrapped exponential
2538:Multivariate stable
2533:Multivariate normal
1854:Benktander 2nd kind
1849:Benktander 1st kind
1638:Discrete phase-type
555:stable distribution
550:normal distribution
163:stable distribution
155:Cauchy distribution
151:normal distribution
2456:Rectified Gaussian
2341:Generalized Pareto
2199:Generalized normal
2071:Matrix-exponential
1326:Lukacs, E. (1970)
1275:
1229:
1183:
1135:
1089:
1051:
654:stochastic process
613:
590:
531:
473:
450:
404:
402:
227:), other than the
167:Gamma distribution
29:probability theory
2867:
2866:
2464:
2463:
2433:
2432:
2324:whose type varies
2270:Normal (Gaussian)
2224:Hyperbolic secant
2173:Exponential power
2076:Maxwell–Boltzmann
1824:Wigner semicircle
1716:
1715:
1688:Parabolic fractal
1678:Negative binomial
1386:978-0-521-55302-5
773:, we can choose (
575:
520:
452:
435:
175:Wald distribution
159:Lévy distribution
16:(Redirected from
2897:
2857:
2856:
2847:
2846:
2786:Compound Poisson
2761:
2749:
2718:von Mises–Fisher
2714:
2702:
2690:
2652:Circular uniform
2648:
2568:
2512:
2483:
2444:
2346:Marchenko–Pastur
2209:Geometric stable
2126:Truncated normal
2019:Inverse Gaussian
1925:Hyperexponential
1764:Beta rectangular
1732:bounded interval
1727:
1595:Discrete uniform
1580:Poisson binomial
1531:
1506:
1499:
1492:
1483:
1478:(Marcel Dekker).
1441:
1440:
1422:
1416:
1415:
1397:
1391:
1390:
1372:
1366:
1365:
1337:
1331:
1324:
1295:Cramér's theorem
1284:
1282:
1281:
1276:
1274:
1273:
1258:
1257:
1238:
1236:
1235:
1230:
1228:
1227:
1212:
1211:
1192:
1190:
1189:
1184:
1182:
1181:
1166:
1165:
1144:
1142:
1141:
1136:
1134:
1133:
1118:
1117:
1098:
1096:
1095:
1090:
1060:
1058:
1057:
1052:
1050:
1049:
1034:
1033:
1014:additive process
1008:Additive process
1002:Additive process
939:, we can define
898: <
708:depends only on
682: <
622:
620:
619:
614:
603:
602:
589:
559:triangular array
540:
538:
537:
532:
521:
518:
501:
497:
496:
472:
449:
413:
411:
410:
405:
403:
377:
376:
365:
364:
353:
352:
339:
338:
327:
326:
313:
312:
223:(≈ finite-sized
126:Bruno de Finetti
45:random variables
21:
2905:
2904:
2900:
2899:
2898:
2896:
2895:
2894:
2870:
2869:
2868:
2863:
2835:
2811:Maximum entropy
2769:
2757:
2745:
2735:
2727:
2710:
2698:
2686:
2641:
2628:
2565:Matrix-valued:
2562:
2508:
2479:
2471:
2460:
2448:
2439:
2429:
2323:
2317:
2234:
2160:
2158:
2152:
2081:Maxwell–Jüttner
1930:Hypoexponential
1836:
1834:
1833:supported on a
1828:
1789:Noncentral beta
1749:Balding–Nichols
1731:
1730:supported on a
1722:
1712:
1615:
1609:
1605:Zipf–Mandelbrot
1535:
1526:
1520:
1510:
1471:6, 57–64.
1450:
1445:
1444:
1437:
1424:
1423:
1419:
1412:
1399:
1398:
1394:
1387:
1374:
1373:
1369:
1339:
1338:
1334:
1325:
1318:
1313:
1291:
1259:
1249:
1241:
1240:
1213:
1203:
1195:
1194:
1167:
1157:
1149:
1148:
1119:
1109:
1101:
1100:
1075:
1074:
1035:
1025:
1017:
1016:
1010:
1004:
985:
976:
966:
956:
947:
930:rational number
914:
893:
884:
876:). Similarly,
875:
857:
848:
835:
821:
814:
801:
792:
782:
768:
750:
737:
728:
707:
698:
678:means that for
661:
646:
640:
594:
570:
569:
485:
481:
430:
429:
401:
400:
395:
390:
385:
379:
378:
368:
366:
356:
354:
344:
341:
340:
330:
328:
318:
315:
314:
304:
295:
294:
285:
276:
266:
254:: the limit as
248:
196:(and hence the
147:
128:. This type of
116:
107:
97:
88:
79:
23:
22:
15:
12:
11:
5:
2903:
2901:
2893:
2892:
2887:
2882:
2872:
2871:
2865:
2864:
2862:
2861:
2851:
2840:
2837:
2836:
2834:
2833:
2828:
2823:
2818:
2813:
2808:
2806:Location–scale
2803:
2798:
2793:
2788:
2783:
2777:
2775:
2771:
2770:
2768:
2767:
2762:
2755:
2750:
2742:
2740:
2729:
2728:
2726:
2725:
2720:
2715:
2708:
2703:
2696:
2691:
2684:
2679:
2674:
2669:
2667:Wrapped Cauchy
2664:
2662:Wrapped normal
2659:
2654:
2649:
2638:
2636:
2630:
2629:
2627:
2626:
2625:
2624:
2619:
2617:Normal-inverse
2614:
2609:
2599:
2598:
2597:
2587:
2579:
2574:
2569:
2560:
2559:
2558:
2548:
2540:
2535:
2530:
2525:
2524:
2523:
2513:
2506:
2505:
2504:
2499:
2489:
2484:
2476:
2474:
2466:
2465:
2462:
2461:
2459:
2458:
2452:
2450:
2441:
2435:
2434:
2431:
2430:
2428:
2427:
2422:
2417:
2409:
2401:
2393:
2384:
2375:
2366:
2357:
2348:
2343:
2338:
2333:
2327:
2325:
2319:
2318:
2316:
2315:
2310:
2308:Variance-gamma
2305:
2300:
2292:
2287:
2282:
2277:
2272:
2267:
2259:
2254:
2253:
2252:
2242:
2237:
2232:
2226:
2221:
2216:
2211:
2206:
2201:
2196:
2188:
2183:
2175:
2170:
2164:
2162:
2154:
2153:
2151:
2150:
2148:Wilks's lambda
2145:
2144:
2143:
2133:
2128:
2123:
2118:
2113:
2108:
2103:
2098:
2093:
2088:
2086:Mittag-Leffler
2083:
2078:
2073:
2068:
2063:
2058:
2053:
2048:
2043:
2038:
2033:
2028:
2027:
2026:
2016:
2007:
2002:
1997:
1996:
1995:
1985:
1983:gamma/Gompertz
1980:
1979:
1978:
1973:
1963:
1958:
1953:
1952:
1951:
1939:
1938:
1937:
1932:
1927:
1917:
1916:
1915:
1905:
1900:
1895:
1894:
1893:
1892:
1891:
1881:
1871:
1866:
1861:
1856:
1851:
1846:
1840:
1838:
1835:semi-infinite
1830:
1829:
1827:
1826:
1821:
1816:
1811:
1806:
1801:
1796:
1791:
1786:
1781:
1776:
1771:
1766:
1761:
1756:
1751:
1746:
1741:
1735:
1733:
1724:
1718:
1717:
1714:
1713:
1711:
1710:
1705:
1700:
1695:
1690:
1685:
1680:
1675:
1670:
1665:
1660:
1655:
1650:
1645:
1640:
1635:
1630:
1625:
1619:
1617:
1614:with infinite
1611:
1610:
1608:
1607:
1602:
1597:
1592:
1587:
1582:
1577:
1576:
1575:
1568:Hypergeometric
1565:
1560:
1555:
1550:
1545:
1539:
1537:
1528:
1522:
1521:
1511:
1509:
1508:
1501:
1494:
1486:
1480:
1479:
1472:
1465:
1449:
1446:
1443:
1442:
1435:
1417:
1410:
1392:
1385:
1367:
1348:(3): 121–148.
1332:
1315:
1314:
1312:
1309:
1308:
1307:
1302:
1297:
1290:
1287:
1272:
1269:
1266:
1262:
1256:
1252:
1248:
1226:
1223:
1220:
1216:
1210:
1206:
1202:
1180:
1177:
1174:
1170:
1164:
1160:
1156:
1132:
1129:
1126:
1122:
1116:
1112:
1108:
1088:
1085:
1082:
1048:
1045:
1042:
1038:
1032:
1028:
1024:
1006:Main article:
1003:
1000:
981:
971:
961:
952:
943:
912:
889:
880:
862:
853:
840:
826:
819:
806:
797:
787:
777:
764:
748:
733:
724:
703:
694:
659:
642:Main article:
639:
636:
624:
623:
612:
609:
606:
601:
597:
593:
588:
585:
582:
578:
542:
541:
530:
527:
524:
516:
513:
510:
507:
504:
500:
495:
492:
488:
484:
480:
477:
471:
468:
465:
462:
459:
455:
448:
445:
442:
438:
415:
414:
399:
396:
394:
391:
389:
386:
384:
381:
380:
375:
371:
367:
363:
359:
355:
351:
347:
343:
342:
337:
333:
329:
325:
321:
317:
316:
311:
307:
303:
302:
281:
271:
262:
247:
244:
146:
143:
112:
102:
93:
84:
74:
66:, there exist
24:
14:
13:
10:
9:
6:
4:
3:
2:
2902:
2891:
2888:
2886:
2883:
2881:
2878:
2877:
2875:
2860:
2852:
2850:
2842:
2841:
2838:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2778:
2776:
2772:
2766:
2763:
2760:
2756:
2754:
2751:
2748:
2744:
2743:
2741:
2739:
2734:
2730:
2724:
2721:
2719:
2716:
2713:
2709:
2707:
2704:
2701:
2697:
2695:
2692:
2689:
2685:
2683:
2680:
2678:
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2658:
2655:
2653:
2650:
2647:
2646:
2640:
2639:
2637:
2635:
2631:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2605:
2604:
2603:
2600:
2596:
2593:
2592:
2591:
2588:
2586:
2585:
2580:
2578:
2577:Matrix normal
2575:
2573:
2570:
2567:
2566:
2561:
2557:
2554:
2553:
2552:
2549:
2547:
2546:
2543:Multivariate
2541:
2539:
2536:
2534:
2531:
2529:
2526:
2522:
2519:
2518:
2517:
2514:
2511:
2507:
2503:
2500:
2498:
2495:
2494:
2493:
2490:
2488:
2485:
2482:
2478:
2477:
2475:
2473:
2470:Multivariate
2467:
2457:
2454:
2453:
2451:
2445:
2442:
2436:
2426:
2423:
2421:
2418:
2416:
2414:
2410:
2408:
2406:
2402:
2400:
2398:
2394:
2392:
2390:
2385:
2383:
2381:
2376:
2374:
2372:
2367:
2365:
2363:
2358:
2356:
2354:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2328:
2326:
2322:with support
2320:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2298:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2265:
2260:
2258:
2255:
2251:
2248:
2247:
2246:
2243:
2241:
2238:
2236:
2235:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2195:
2194:
2189:
2187:
2184:
2182:
2181:
2176:
2174:
2171:
2169:
2166:
2165:
2163:
2159:on the whole
2155:
2149:
2146:
2142:
2139:
2138:
2137:
2134:
2132:
2131:type-2 Gumbel
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
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:
2025:
2022:
2021:
2020:
2017:
2015:
2013:
2008:
2006:
2003:
2001:
2000:Half-logistic
1998:
1994:
1991:
1990:
1989:
1986:
1984:
1981:
1977:
1974:
1972:
1969:
1968:
1967:
1964:
1962:
1959:
1957:
1956:Folded normal
1954:
1950:
1947:
1946:
1945:
1944:
1940:
1936:
1933:
1931:
1928:
1926:
1923:
1922:
1921:
1918:
1914:
1911:
1910:
1909:
1906:
1904:
1901:
1899:
1896:
1890:
1887:
1886:
1885:
1882:
1880:
1877:
1876:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1841:
1839:
1831:
1825:
1822:
1820:
1817:
1815:
1812:
1810:
1807:
1805:
1802:
1800:
1799:Raised cosine
1797:
1795:
1792:
1790:
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1770:
1767:
1765:
1762:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1736:
1734:
1728:
1725:
1719:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1673:Mixed Poisson
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1644:
1641:
1639:
1636:
1634:
1631:
1629:
1626:
1624:
1621:
1620:
1618:
1612:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1574:
1571:
1570:
1569:
1566:
1564:
1561:
1559:
1556:
1554:
1553:Beta-binomial
1551:
1549:
1546:
1544:
1541:
1540:
1538:
1532:
1529:
1523:
1518:
1514:
1507:
1502:
1500:
1495:
1493:
1488:
1487:
1484:
1477:
1473:
1470:
1466:
1464:
1460:
1456:
1452:
1451:
1447:
1438:
1436:9780521553025
1432:
1428:
1421:
1418:
1413:
1411:0-471-58494-0
1407:
1403:
1396:
1393:
1388:
1382:
1378:
1371:
1368:
1363:
1359:
1355:
1351:
1347:
1343:
1336:
1333:
1329:
1323:
1321:
1317:
1310:
1306:
1303:
1301:
1298:
1296:
1293:
1292:
1288:
1286:
1270:
1267:
1264:
1254:
1250:
1224:
1221:
1218:
1208:
1204:
1178:
1175:
1172:
1162:
1158:
1146:
1130:
1127:
1124:
1114:
1110:
1086:
1083:
1080:
1072:
1068:
1064:
1046:
1043:
1040:
1030:
1026:
1015:
1009:
1001:
999:
997:
994: −
993:
989:
984:
980:
974:
970:
964:
960:
955:
951:
946:
942:
938:
934:
931:
927:
924: −
923:
919:
916: :
915:
908:
903:
901:
897:
892:
888:
883:
879:
874:
870:
866:
861:
856:
852:
847:
843:
839:
834:
830:
825:
818:
813:
809:
805:
800:
796:
790:
786:
780:
776:
772:
767:
763:
759:
755:
752: :
751:
743:
741:
736:
732:
727:
723:
719:
715:
712: −
711:
706:
702:
697:
693:
689:
685:
681:
677:
673:
670:
666:
663: :
662:
655:
651:
645:
637:
635:
633:
629:
610:
607:
604:
599:
595:
591:
580:
568:
567:
566:
564:
560:
556:
551:
547:
528:
525:
522:
514:
511:
505:
502:
498:
493:
490:
486:
482:
475:
469:
466:
463:
460:
457:
440:
428:
427:
426:
424:
420:
397:
392:
387:
382:
373:
369:
361:
357:
349:
345:
335:
331:
323:
319:
309:
305:
293:
292:
291:
289:
284:
280:
274:
270:
265:
261:
257:
253:
246:Limit theorem
245:
243:
241:
236:
234:
230:
226:
222:
218:
214:
210:
205:
203:
199:
195:
191:
186:
184:
180:
176:
172:
168:
164:
160:
156:
152:
144:
142:
139:
135:
131:
127:
122:
120:
115:
111:
105:
101:
96:
92:
87:
83:
77:
73:
69:
65:
61:
56:
54:
50:
46:
42:
38:
34:
30:
19:
2758:
2746:
2712:Multivariate
2711:
2699:
2687:
2682:Wrapped Lévy
2642:
2590:Matrix gamma
2583:
2563:
2551:Normal-gamma
2544:
2510:Continuous:
2509:
2480:
2425:Tukey lambda
2412:
2404:
2399:-exponential
2396:
2388:
2379:
2370:
2361:
2355:-exponential
2352:
2296:
2263:
2230:
2192:
2179:
2106:Poly-Weibull
2051:Log-logistic
2011:
2010:Hotelling's
1942:
1784:Logit-normal
1658:Gauss–Kuzmin
1653:Flory–Schulz
1534:with finite
1475:
1468:
1454:
1426:
1420:
1401:
1395:
1376:
1370:
1345:
1341:
1335:
1327:
1147:
1011:
995:
991:
982:
978:
972:
968:
962:
958:
953:
949:
944:
940:
936:
932:
925:
921:
917:
910:
906:
904:
899:
895:
890:
886:
881:
877:
872:
868:
864:
859:
854:
850:
845:
841:
837:
832:
828:
823:
816:
811:
807:
803:
798:
794:
788:
784:
778:
774:
770:
765:
761:
757:
753:
746:
744:
734:
730:
725:
721:
717:
713:
709:
704:
700:
695:
691:
683:
679:
675:
664:
657:
650:Lévy process
647:
644:Lévy process
638:Lévy process
627:
625:
543:
422:
416:
282:
278:
272:
268:
263:
259:
255:
249:
237:
216:
206:
187:
148:
123:
118:
113:
109:
103:
99:
94:
90:
85:
81:
75:
71:
67:
63:
59:
57:
52:
36:
26:
2796:Exponential
2645:directional
2634:Directional
2521:Generalized
2492:Multinomial
2447:continuous-
2387:Kaniadakis
2378:Kaniadakis
2369:Kaniadakis
2360:Kaniadakis
2351:Kaniadakis
2303:Tracy–Widom
2280:Skew normal
2262:Noncentral
2046:Log-Laplace
2024:Generalized
2005:Half-normal
1971:Generalized
1935:Logarithmic
1920:Exponential
1874:Chi-squared
1814:U-quadratic
1779:Kumaraswamy
1721:Continuous
1668:Logarithmic
1563:Categorical
740:independent
288:independent
200:also). The
134:probability
132:is used in
2874:Categories
2791:Elliptical
2747:Degenerate
2733:Degenerate
2481:Discrete:
2440:univariate
2295:Student's
2250:Asymmetric
2229:Johnson's
2157:supported
2101:Phase-type
2056:Log-normal
2041:Log-Cauchy
2031:Kolmogorov
1949:Noncentral
1879:Noncentral
1859:Beta prime
1809:Triangular
1804:Reciprocal
1774:Irwin–Hall
1723:univariate
1703:Yule–Simon
1585:Rademacher
1527:univariate
1448:References
990:values of
988:Irrational
745:If {
716:and where
676:stationary
669:stationary
419:weak sense
233:reciprocal
138:statistics
89:whose sum
2516:Dirichlet
2497:Dirichlet
2407:-Gaussian
2382:-Logistic
2219:Holtsmark
2191:Gaussian
2178:Fisher's
2161:real line
1663:Geometric
1643:Delaporte
1548:Bernoulli
1525:Discrete
1362:0346-1238
1311:Footnotes
1268:≥
1251:μ
1222:≥
1205:μ
1176:≥
1159:μ
1128:≥
1111:μ
1084:≥
1044:≥
608:λ
587:∞
584:→
523:ε
506:ε
467:≤
461:≤
447:∞
444:→
398:⋱
393:⋮
388:⋮
383:⋮
43:(i.i.d.)
2849:Category
2781:Circular
2774:Families
2759:Singular
2738:singular
2502:Negative
2449:discrete
2415:-Weibull
2373:-Weibull
2257:Logistic
2141:Discrete
2111:Rayleigh
2091:Nakagami
2014:-squared
1988:Gompertz
1837:interval
1573:Negative
1558:Binomial
1289:See also
977:+ ... +
674:, where
546:variance
277:+ ... +
211:and the
192:and the
181:and the
145:Examples
108:+ ... +
2859:Commons
2831:Wrapped
2826:Tweedie
2821:Pearson
2816:Mixture
2723:Bingham
2622:Complex
2612:Inverse
2602:Wishart
2595:Inverse
2582:Matrix
2556:Inverse
2472:(joint)
2391:-Erlang
2245:Laplace
2136:Weibull
1993:Shifted
1976:Inverse
1961:Fréchet
1884:Inverse
1819:Uniform
1739:Arcsine
1698:Skellam
1693:Poisson
1616:support
1590:Soliton
1543:Benford
1536:support
849:, ...,
793:, ...,
656:{
221:support
80:, ...,
2765:Cantor
2607:Normal
2438:Mixed
2364:-Gamma
2290:Stable
2240:Landau
2214:Gumbel
2168:Cauchy
2096:Pareto
1908:Erlang
1889:Scaled
1844:Benini
1683:Panjer
1433:
1408:
1383:
1360:
1099:. Let
1063:cadlag
686:, the
225:domain
177:, the
173:, the
169:, the
157:, the
153:, the
47:. The
2487:Ewens
2313:Voigt
2285:Slash
2066:Lomax
2061:Log-t
1966:Gamma
1913:Hyper
1903:Davis
1898:Dagum
1754:Bates
1744:ARGUS
1628:Borel
802:) = (
2736:and
2694:Kent
2121:Rice
2036:Lévy
1864:Burr
1794:PERT
1759:Beta
1708:Zeta
1600:Zipf
1517:list
1431:ISBN
1406:ISBN
1381:ISBN
1358:ISSN
1346:1977
526:>
503:>
238:Any
215:are
207:The
136:and
31:, a
2572:LKJ
1869:Chi
1459:doi
1350:doi
1061:(a
1012:An
986:.
867:−1)
738:is
690:of
577:lim
454:max
437:lim
286:of
217:not
35:is
27:In
2876::
1356:.
1344:.
1319:^
1065:,
983:qp
967:+
948:−
902:.
885:−
858:−
836:−
822:,
815:−
799:nn
783:,
729:−
699:−
634:.
529:0.
374:33
362:32
350:31
336:22
324:21
310:11
283:nn
267:=
185:.
121:.
114:nn
98:=
86:nn
55:.
2584:t
2545:t
2413:q
2405:q
2397:q
2389:κ
2380:κ
2371:κ
2362:κ
2353:κ
2297:t
2264:t
2233:U
2231:S
2193:q
2180:z
2012:T
1943:F
1519:)
1515:(
1505:e
1498:t
1491:v
1461::
1439:.
1414:.
1389:.
1364:.
1352::
1271:0
1265:t
1261:}
1255:t
1247:{
1225:0
1219:t
1215:}
1209:t
1201:{
1179:0
1173:t
1169:}
1163:t
1155:{
1131:0
1125:t
1121:}
1115:t
1107:{
1087:0
1081:t
1047:0
1041:t
1037:}
1031:t
1027:X
1023:{
996:s
992:t
979:X
975:2
973:q
969:X
965:1
963:q
959:X
954:s
950:L
945:t
941:L
937:q
935:/
933:p
926:s
922:t
918:t
913:t
911:L
907:F
900:t
896:s
891:s
887:L
882:t
878:L
873:n
871:/
869:t
865:n
863:(
860:L
855:t
851:L
846:n
844:/
842:t
838:L
833:n
831:/
829:t
827:2
824:L
820:0
817:L
812:n
810:/
808:t
804:L
795:X
791:2
789:n
785:X
781:1
779:n
775:X
771:n
766:t
762:L
758:t
754:t
749:t
747:L
735:s
731:L
726:t
722:L
714:s
710:t
705:s
701:L
696:t
692:L
684:t
680:s
665:t
660:t
658:L
628:λ
611:,
605:=
600:n
596:p
592:n
581:n
515:0
512:=
509:)
499:|
494:k
491:n
487:X
483:|
479:(
476:P
470:n
464:k
458:1
441:n
370:X
358:X
346:X
332:X
320:X
306:X
279:X
275:1
273:n
269:X
264:n
260:S
256:n
119:F
110:X
106:1
104:n
100:X
95:n
91:S
82:X
78:1
76:n
72:X
68:n
64:n
60:F
20:)
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