22:
2062:
2272:
2487:
270:
967:
177:
1906:
1668:
1107:
1542:
698:
409:
2305:
816:
774:
1163:
1475:
1285:
874:
1725:
1561:
50:
2178:
2087:
1384:
1042:
894:
836:
725:
1435:
2514:
2354:
2173:
1193:
629:
579:
300:
1830:
1748:
1588:
1213:
987:
599:
456:
328:
93:
2549:
2146:
1305:
352:
1882:
1786:
1620:
1337:
1245:
1019:
519:
492:
2359:
2103:
896:
is positive for all the non-negative polynomials in the univariate case. By
Haviland's theorem, the linear functional has a measure form, that is
2081:
2670:
2069:
188:
899:
104:
2840:
2783:
2748:
2704:
876:. In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional
1833:
1884:. For the problem on an infinite interval, uniqueness is a more delicate question. There are distributions, such as
2835:
2057:{\displaystyle d\mu (x)=\rho (x)dx,\quad \rho (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x)m_{n}}
2315:, we know that the standard normal distribution is a determinate measure, thus we have the following form of the
2830:
2810:
1888:, which have finite moments for all the positive integers but where other distributions have the same moments.
1628:
1050:
2825:
2820:
1480:
637:
357:
2591:
779:
730:
529:
2277:
1118:
1445:
1363:
1255:
841:
2663:
2735:. Translations of Mathematical Monographs. Providence, Rhode Island: American Mathematical Society.
2586:
2576:
2566:
1387:
704:
459:
431:
1676:
2815:
2561:
2312:
1885:
1803:
465:
28:
1369:
1027:
879:
821:
710:
2316:
1392:
435:
73:
2581:
2118:
The moment problem has applications to probability theory. The following is commonly used:
1897:
1849:
1807:
1171:
707:. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional
607:
557:
278:
96:
2492:
2332:
2151:
1815:
1733:
1573:
1198:
972:
584:
441:
313:
78:
8:
2770:. Graduate Texts in Mathematics. Vol. 277. Cham: Springer International Publishing.
2148:
is a determinate measure (i.e. its moments determine it uniquely), and the measures
969:. A condition of similar form is necessary and sufficient for the existence of a measure
416:
57:
2519:
2719:
2107:
1290:
412:
337:
2131:
1855:
1759:
1593:
1310:
1218:
992:
2789:
2779:
2754:
2744:
2700:
533:
2771:
2736:
525:
2267:{\displaystyle \forall k\geq 0\quad \lim _{n\rightarrow \infty }m_{k}\left=m_{k},}
602:
2692:
2775:
1896:
When the solution exists, it can be formally written using derivatives of the
2804:
2793:
2758:
2714:
2571:
2068:
The expression can be derived by taking the inverse
Fourier transform of its
21:
1845:
543:
2102:). Results on the truncated moment problem have numerous applications to
65:
2740:
601:
if and only if a certain positivity condition is fulfilled; namely, the
1837:
2482:{\displaystyle m_{2k}\to {\frac {(2k)!}{2^{k}k!}};\quad m_{2k+1}\to 0}
1841:
331:
265:{\displaystyle m_{n}=\int _{-\infty }^{\infty }M_{n}(x)\,d\mu (x)\,.}
1024:
One way to prove these results is to consider the linear functional
962:{\displaystyle \Lambda (x^{n})=\int _{-\infty }^{\infty }x^{n}d\mu }
2721:
The classical moment problem and some related questions in analysis
424:
53:
172:{\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}\,d\mu (x)\,.}
72:
arises as the result of trying to invert the mapping that takes a
1756:). Using a representation theorem for positive polynomials on
2633:
2631:
2090:, which studies the properties of measures with fixed first
420:
2628:
2616:
818:(non-negative for sum of squares of polynomials). Assume
2604:
2522:
2495:
2335:
2280:
2154:
2134:
18:
Trying to map moments to a measure that generates them
2362:
2181:
1909:
1858:
1818:
1762:
1736:
1679:
1631:
1596:
1576:
1483:
1448:
1395:
1372:
1313:
1293:
1258:
1221:
1201:
1174:
1121:
1053:
1030:
995:
975:
902:
882:
844:
824:
782:
733:
713:
640:
610:
587:
560:
495:
468:
444:
430:
There are three named classical moment problems: the
360:
340:
316:
281:
191:
107:
81:
31:
2643:
2718:
2543:
2508:
2481:
2348:
2299:
2266:
2167:
2140:
2056:
1876:
1824:
1780:
1742:
1719:
1662:
1614:
1582:
1536:
1469:
1429:
1388:space of continuous functions with compact support
1378:
1331:
1299:
1279:
1239:
1207:
1187:
1157:
1101:
1036:
1013:
981:
961:
888:
868:
830:
810:
768:
719:
692:
623:
593:
573:
513:
486:
450:
403:
346:
322:
294:
264:
171:
87:
44:
1832:in the Hausdorff moment problem follows from the
2802:
2196:
2733:The Markov Moment Problem and Extremal Problems
60:is the distribution solving the moment problem.
2730:
2690:
2637:
2622:
1797:
532:in which the Hankel matrices are replaced by
398:
361:
2699:. New York: American mathematical society.
2329:If a sequence of probability distributions
2727:(translated from the Russian by N. Kemmer)
1524:
458:is allowed to be the whole real line; the
2765:
2610:
1663:{\displaystyle \varphi (f)=\int f\,d\mu }
1653:
846:
686:
258:
242:
165:
149:
2713:
2649:
1102:{\displaystyle P(x)=\sum _{k}a_{k}x^{k}}
581:is the sequence of moments of a measure
20:
2731:KreÄn, M. G.; Nudel′man, A. A. (1977).
2082:Chebyshev–Markov–Stieltjes inequalities
1537:{\displaystyle f\in C_{c}(),\;f\geq 0.}
693:{\displaystyle (H_{n})_{ij}=m_{i+j}\,,}
411:. In this form the question appears in
404:{\displaystyle \{x^{n}:n=1,2,\dotsc \}}
275:for an arbitrary sequence of functions
2803:
2661:
2300:{\textstyle \mu _{n}\rightarrow \mu }
2106:, optimisation and limit theorems in
1794:) as a condition on Hankel matrices.
811:{\displaystyle \Lambda (f^{2})\geq 0}
769:{\displaystyle \Lambda (x^{n})=m_{n}}
427:and so on, and whether it is unique.
25:Example: Given the mean and variance
1439:
1249:
1158:{\displaystyle \sum _{k}a_{k}m_{k}.}
1570:) holds iff there exists a measure
524:The moment problem also extends to
13:
2206:
2182:
1981:
1891:
1730:Thus the existence of the measure
938:
933:
903:
883:
825:
783:
734:
714:
478:
218:
213:
134:
129:
14:
2852:
2725:. New York: Hafner Publishing Co.
1834:Weierstrass approximation theorem
1470:{\displaystyle \varphi (f)\geq 0}
1280:{\displaystyle \varphi (P)\geq 0}
182:More generally, one may consider
2676:from the original on 1 Jul 2022.
1195:are the moments of some measure
869:{\displaystyle \mathbb {R} ^{*}}
2437:
2194:
1949:
305:
2664:"The classical moment problem"
2662:Sodin, Sasha (March 5, 2019).
2655:
2538:
2526:
2473:
2470:
2457:
2407:
2398:
2392:
2389:
2376:
2291:
2258:
2252:
2203:
2113:
2086:An important variation is the
2041:
2035:
2030:
2024:
1999:
1989:
1959:
1953:
1937:
1931:
1922:
1916:
1871:
1859:
1775:
1763:
1714:
1711:
1699:
1696:
1641:
1635:
1609:
1597:
1518:
1515:
1503:
1500:
1458:
1452:
1424:
1421:
1409:
1406:
1326:
1314:
1268:
1262:
1234:
1222:
1063:
1057:
1008:
996:
989:supported on a given interval
919:
906:
857:
850:
799:
786:
750:
737:
655:
641:
508:
496:
481:
469:
255:
249:
239:
233:
162:
156:
1:
2841:Optimization in vector spaces
2684:
2075:
415:, asking whether there is a
2592:Trigonometric moment problem
1720:{\displaystyle f\in C_{c}()}
1562:Riesz representation theorem
549:
530:trigonometric moment problem
7:
2555:
1798:Uniqueness (or determinacy)
1790:
1752:
1566:
1550:
1362:) holds, one can apply the
1358:
1346:
487:{\displaystyle [0,\infty )}
45:{\displaystyle \sigma ^{2}}
10:
2857:
2766:SchmĂĽdgen, Konrad (2017).
2638:KreÄn & Nudel′man 1977
2623:Shohat & Tamarkin 1943
2079:
1801:
1364:M. Riesz extension theorem
546:instead of the real line.
310:In the classical setting,
2776:10.1007/978-3-319-64546-9
2691:Shohat, James Alexander;
52:(as well as all further
2597:
2587:Stieltjes moment problem
2577:Hausdorff moment problem
2567:Hamburger moment problem
2123:Theorem (Fréchet-Shohat)
2088:truncated moment problem
1886:log-normal distributions
1788:, one can reformulate (
1379:{\displaystyle \varphi }
1307:that is non-negative on
1044:that sends a polynomial
1037:{\displaystyle \varphi }
889:{\displaystyle \Lambda }
831:{\displaystyle \Lambda }
720:{\displaystyle \Lambda }
460:Stieltjes moment problem
432:Hamburger moment problem
2836:Real algebraic geometry
2070:characteristic function
1430:{\displaystyle C_{c}()}
1386:to a functional on the
2697:The Problem of Moments
2545:
2510:
2483:
2350:
2301:
2268:
2169:
2142:
2096:moments (for a finite
2058:
1985:
1878:
1826:
1782:
1744:
1721:
1664:
1616:
1584:
1538:
1471:
1431:
1380:
1333:
1301:
1281:
1241:
1209:
1189:
1159:
1103:
1038:
1015:
983:
963:
890:
870:
832:
812:
770:
721:
705:positive semi-definite
694:
625:
595:
575:
554:A sequence of numbers
515:
488:
452:
405:
348:
324:
296:
266:
173:
89:
61:
46:
2831:Mathematical problems
2811:Mathematical analysis
2546:
2511:
2509:{\textstyle \nu _{n}}
2484:
2351:
2349:{\textstyle \nu _{n}}
2317:central limit theorem
2302:
2269:
2170:
2168:{\textstyle \mu _{n}}
2143:
2059:
1965:
1879:
1827:
1783:
1745:
1722:
1665:
1617:
1585:
1539:
1472:
1432:
1381:
1334:
1302:
1282:
1242:
1210:
1190:
1188:{\displaystyle m_{k}}
1160:
1104:
1039:
1016:
984:
964:
891:
871:
833:
813:
771:
722:
695:
626:
624:{\displaystyle H_{n}}
596:
576:
574:{\displaystyle m_{n}}
516:
489:
453:
406:
349:
325:
297:
295:{\displaystyle M_{n}}
267:
174:
90:
47:
24:
2826:Moment (mathematics)
2821:Probability problems
2582:Moment (mathematics)
2562:Carleman's condition
2520:
2493:
2360:
2333:
2313:Carleman's condition
2278:
2179:
2152:
2132:
1907:
1898:Dirac delta function
1856:
1850:continuous functions
1836:, which states that
1825:{\displaystyle \mu }
1816:
1804:Carleman's condition
1760:
1743:{\displaystyle \mu }
1734:
1677:
1629:
1594:
1583:{\displaystyle \mu }
1574:
1481:
1446:
1393:
1370:
1311:
1291:
1256:
1219:
1208:{\displaystyle \mu }
1199:
1172:
1119:
1051:
1028:
993:
982:{\displaystyle \mu }
973:
900:
880:
842:
838:can be extended to
822:
780:
731:
711:
638:
608:
594:{\displaystyle \mu }
585:
558:
493:
466:
451:{\displaystyle \mu }
442:
358:
338:
330:is a measure on the
323:{\displaystyle \mu }
314:
279:
189:
105:
88:{\displaystyle \mu }
79:
29:
2544:{\textstyle N(0,1)}
2327: —
2126: —
1287:for any polynomial
942:
544:complex unit circle
536:and the support of
417:probability measure
222:
138:
95:to the sequence of
58:normal distribution
2768:The Moment Problem
2693:Tamarkin, Jacob D.
2541:
2506:
2479:
2346:
2325:
2297:
2264:
2210:
2165:
2138:
2124:
2108:probability theory
2054:
1874:
1822:
1812:The uniqueness of
1778:
1750:is equivalent to (
1740:
1717:
1660:
1612:
1580:
1534:
1467:
1427:
1376:
1329:
1297:
1277:
1237:
1205:
1185:
1155:
1131:
1099:
1078:
1034:
1011:
979:
959:
925:
886:
866:
828:
808:
766:
717:
690:
621:
591:
571:
511:
484:
448:
413:probability theory
401:
344:
320:
292:
262:
205:
169:
121:
85:
62:
42:
2785:978-3-319-64545-2
2750:978-0-8218-4500-4
2741:10.1090/mmono/050
2715:Akhiezer, Naum I.
2706:978-1-4704-1228-9
2551:in distribution.
2432:
2323:
2307:in distribution.
2195:
2141:{\textstyle \mu }
2122:
2104:extremal problems
2017:
1808:Krein's condition
1558:
1557:
1354:
1353:
1300:{\displaystyle P}
1247:, then evidently
1122:
1069:
534:Toeplitz matrices
419:having specified
347:{\displaystyle M}
2848:
2797:
2762:
2726:
2724:
2710:
2678:
2677:
2675:
2668:
2659:
2653:
2647:
2641:
2635:
2626:
2620:
2614:
2608:
2550:
2548:
2547:
2542:
2515:
2513:
2512:
2507:
2505:
2504:
2488:
2486:
2485:
2480:
2469:
2468:
2456:
2455:
2433:
2431:
2424:
2423:
2413:
2396:
2388:
2387:
2375:
2374:
2355:
2353:
2352:
2347:
2345:
2344:
2328:
2306:
2304:
2303:
2298:
2290:
2289:
2273:
2271:
2270:
2265:
2251:
2250:
2238:
2234:
2233:
2220:
2219:
2209:
2174:
2172:
2171:
2166:
2164:
2163:
2147:
2145:
2144:
2139:
2127:
2101:
2095:
2063:
2061:
2060:
2055:
2053:
2052:
2034:
2033:
2018:
2016:
2008:
2007:
2006:
1987:
1984:
1979:
1883:
1881:
1880:
1877:{\displaystyle }
1875:
1848:in the space of
1831:
1829:
1828:
1823:
1787:
1785:
1784:
1781:{\displaystyle }
1779:
1749:
1747:
1746:
1741:
1726:
1724:
1723:
1718:
1695:
1694:
1669:
1667:
1666:
1661:
1621:
1619:
1618:
1615:{\displaystyle }
1613:
1589:
1587:
1586:
1581:
1552:
1543:
1541:
1540:
1535:
1499:
1498:
1476:
1474:
1473:
1468:
1440:
1436:
1434:
1433:
1428:
1405:
1404:
1385:
1383:
1382:
1377:
1356:Vice versa, if (
1348:
1338:
1336:
1335:
1332:{\displaystyle }
1330:
1306:
1304:
1303:
1298:
1286:
1284:
1283:
1278:
1250:
1246:
1244:
1243:
1240:{\displaystyle }
1238:
1214:
1212:
1211:
1206:
1194:
1192:
1191:
1186:
1184:
1183:
1164:
1162:
1161:
1156:
1151:
1150:
1141:
1140:
1130:
1108:
1106:
1105:
1100:
1098:
1097:
1088:
1087:
1077:
1043:
1041:
1040:
1035:
1020:
1018:
1017:
1014:{\displaystyle }
1012:
988:
986:
985:
980:
968:
966:
965:
960:
952:
951:
941:
936:
918:
917:
895:
893:
892:
887:
875:
873:
872:
867:
865:
864:
849:
837:
835:
834:
829:
817:
815:
814:
809:
798:
797:
775:
773:
772:
767:
765:
764:
749:
748:
726:
724:
723:
718:
699:
697:
696:
691:
685:
684:
666:
665:
653:
652:
630:
628:
627:
622:
620:
619:
600:
598:
597:
592:
580:
578:
577:
572:
570:
569:
541:
526:complex analysis
520:
518:
517:
514:{\displaystyle }
512:
491:
490:
485:
457:
455:
454:
449:
410:
408:
407:
402:
373:
372:
354:is the sequence
353:
351:
350:
345:
329:
327:
326:
321:
301:
299:
298:
293:
291:
290:
271:
269:
268:
263:
232:
231:
221:
216:
201:
200:
178:
176:
175:
170:
148:
147:
137:
132:
117:
116:
94:
92:
91:
86:
51:
49:
48:
43:
41:
40:
2856:
2855:
2851:
2850:
2849:
2847:
2846:
2845:
2801:
2800:
2786:
2751:
2707:
2687:
2682:
2681:
2673:
2666:
2660:
2656:
2648:
2644:
2636:
2629:
2621:
2617:
2609:
2605:
2600:
2558:
2553:
2521:
2518:
2517:
2500:
2496:
2494:
2491:
2490:
2464:
2460:
2442:
2438:
2419:
2415:
2414:
2397:
2395:
2383:
2379:
2367:
2363:
2361:
2358:
2357:
2340:
2336:
2334:
2331:
2330:
2326:
2309:
2285:
2281:
2279:
2276:
2275:
2246:
2242:
2229:
2225:
2221:
2215:
2211:
2199:
2180:
2177:
2176:
2159:
2155:
2153:
2150:
2149:
2133:
2130:
2129:
2125:
2116:
2097:
2091:
2084:
2078:
2048:
2044:
2023:
2019:
2009:
2002:
1998:
1988:
1986:
1980:
1969:
1908:
1905:
1904:
1894:
1892:Formal solution
1857:
1854:
1853:
1817:
1814:
1813:
1810:
1800:
1761:
1758:
1757:
1735:
1732:
1731:
1690:
1686:
1678:
1675:
1674:
1630:
1627:
1626:
1595:
1592:
1591:
1575:
1572:
1571:
1494:
1490:
1482:
1479:
1478:
1447:
1444:
1443:
1400:
1396:
1394:
1391:
1390:
1371:
1368:
1367:
1312:
1309:
1308:
1292:
1289:
1288:
1257:
1254:
1253:
1220:
1217:
1216:
1200:
1197:
1196:
1179:
1175:
1173:
1170:
1169:
1146:
1142:
1136:
1132:
1126:
1120:
1117:
1116:
1093:
1089:
1083:
1079:
1073:
1052:
1049:
1048:
1029:
1026:
1025:
994:
991:
990:
974:
971:
970:
947:
943:
937:
929:
913:
909:
901:
898:
897:
881:
878:
877:
860:
856:
845:
843:
840:
839:
823:
820:
819:
793:
789:
781:
778:
777:
760:
756:
744:
740:
732:
729:
728:
712:
709:
708:
674:
670:
658:
654:
648:
644:
639:
636:
635:
615:
611:
609:
606:
605:
603:Hankel matrices
586:
583:
582:
565:
561:
559:
556:
555:
552:
537:
494:
467:
464:
463:
443:
440:
439:
368:
364:
359:
356:
355:
339:
336:
335:
315:
312:
311:
308:
286:
282:
280:
277:
276:
227:
223:
217:
209:
196:
192:
190:
187:
186:
143:
139:
133:
125:
112:
108:
106:
103:
102:
80:
77:
76:
36:
32:
30:
27:
26:
19:
12:
11:
5:
2854:
2844:
2843:
2838:
2833:
2828:
2823:
2818:
2816:Hilbert spaces
2813:
2799:
2798:
2784:
2763:
2749:
2728:
2711:
2705:
2686:
2683:
2680:
2679:
2654:
2642:
2627:
2615:
2613:, p. 257.
2611:SchmĂĽdgen 2017
2602:
2601:
2599:
2596:
2595:
2594:
2589:
2584:
2579:
2574:
2569:
2564:
2557:
2554:
2540:
2537:
2534:
2531:
2528:
2525:
2503:
2499:
2478:
2475:
2472:
2467:
2463:
2459:
2454:
2451:
2448:
2445:
2441:
2436:
2430:
2427:
2422:
2418:
2412:
2409:
2406:
2403:
2400:
2394:
2391:
2386:
2382:
2378:
2373:
2370:
2366:
2343:
2339:
2321:
2296:
2293:
2288:
2284:
2263:
2260:
2257:
2254:
2249:
2245:
2241:
2237:
2232:
2228:
2224:
2218:
2214:
2208:
2205:
2202:
2198:
2193:
2190:
2187:
2184:
2175:are such that
2162:
2158:
2137:
2120:
2115:
2112:
2077:
2074:
2066:
2065:
2051:
2047:
2043:
2040:
2037:
2032:
2029:
2026:
2022:
2015:
2012:
2005:
2001:
1997:
1994:
1991:
1983:
1978:
1975:
1972:
1968:
1964:
1961:
1958:
1955:
1952:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1893:
1890:
1873:
1870:
1867:
1864:
1861:
1821:
1799:
1796:
1777:
1774:
1771:
1768:
1765:
1739:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1693:
1689:
1685:
1682:
1671:
1670:
1659:
1656:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1611:
1608:
1605:
1602:
1599:
1579:
1556:
1555:
1546:
1544:
1533:
1530:
1527:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1497:
1493:
1489:
1486:
1466:
1463:
1460:
1457:
1454:
1451:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1403:
1399:
1375:
1352:
1351:
1342:
1340:
1328:
1325:
1322:
1319:
1316:
1296:
1276:
1273:
1270:
1267:
1264:
1261:
1236:
1233:
1230:
1227:
1224:
1204:
1182:
1178:
1166:
1165:
1154:
1149:
1145:
1139:
1135:
1129:
1125:
1110:
1109:
1096:
1092:
1086:
1082:
1076:
1072:
1068:
1065:
1062:
1059:
1056:
1033:
1010:
1007:
1004:
1001:
998:
978:
958:
955:
950:
946:
940:
935:
932:
928:
924:
921:
916:
912:
908:
905:
885:
863:
859:
855:
852:
848:
827:
807:
804:
801:
796:
792:
788:
785:
763:
759:
755:
752:
747:
743:
739:
736:
716:
701:
700:
689:
683:
680:
677:
673:
669:
664:
661:
657:
651:
647:
643:
618:
614:
590:
568:
564:
551:
548:
510:
507:
504:
501:
498:
483:
480:
477:
474:
471:
447:
400:
397:
394:
391:
388:
385:
382:
379:
376:
371:
367:
363:
343:
319:
307:
304:
289:
285:
273:
272:
261:
257:
254:
251:
248:
245:
241:
238:
235:
230:
226:
220:
215:
212:
208:
204:
199:
195:
180:
179:
168:
164:
161:
158:
155:
152:
146:
142:
136:
131:
128:
124:
120:
115:
111:
84:
70:moment problem
39:
35:
17:
9:
6:
4:
3:
2:
2853:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2808:
2806:
2795:
2791:
2787:
2781:
2777:
2773:
2769:
2764:
2760:
2756:
2752:
2746:
2742:
2738:
2734:
2729:
2723:
2722:
2716:
2712:
2708:
2702:
2698:
2694:
2689:
2688:
2672:
2665:
2658:
2651:
2650:Akhiezer 1965
2646:
2639:
2634:
2632:
2624:
2619:
2612:
2607:
2603:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2572:Hankel matrix
2570:
2568:
2565:
2563:
2560:
2559:
2552:
2535:
2532:
2529:
2523:
2516:converges to
2501:
2497:
2476:
2465:
2461:
2452:
2449:
2446:
2443:
2439:
2434:
2428:
2425:
2420:
2416:
2410:
2404:
2401:
2384:
2380:
2371:
2368:
2364:
2341:
2337:
2320:
2318:
2314:
2308:
2294:
2286:
2282:
2261:
2255:
2247:
2243:
2239:
2235:
2230:
2226:
2222:
2216:
2212:
2200:
2191:
2188:
2185:
2160:
2156:
2135:
2119:
2111:
2109:
2105:
2100:
2094:
2089:
2083:
2073:
2071:
2049:
2045:
2038:
2027:
2020:
2013:
2010:
2003:
1995:
1992:
1976:
1973:
1970:
1966:
1962:
1956:
1950:
1946:
1943:
1940:
1934:
1928:
1925:
1919:
1913:
1910:
1903:
1902:
1901:
1899:
1889:
1887:
1868:
1865:
1862:
1851:
1847:
1843:
1839:
1835:
1819:
1809:
1805:
1795:
1793:
1792:
1772:
1769:
1766:
1755:
1754:
1737:
1728:
1708:
1705:
1702:
1691:
1687:
1683:
1680:
1657:
1654:
1650:
1647:
1644:
1638:
1632:
1625:
1624:
1623:
1606:
1603:
1600:
1590:supported on
1577:
1569:
1568:
1563:
1554:
1547:
1545:
1531:
1528:
1525:
1521:
1512:
1509:
1506:
1495:
1491:
1487:
1484:
1464:
1461:
1455:
1449:
1442:
1441:
1438:
1418:
1415:
1412:
1401:
1397:
1389:
1373:
1365:
1361:
1360:
1350:
1343:
1341:
1323:
1320:
1317:
1294:
1274:
1271:
1265:
1259:
1252:
1251:
1248:
1231:
1228:
1225:
1215:supported on
1202:
1180:
1176:
1152:
1147:
1143:
1137:
1133:
1127:
1123:
1115:
1114:
1113:
1094:
1090:
1084:
1080:
1074:
1070:
1066:
1060:
1054:
1047:
1046:
1045:
1031:
1022:
1005:
1002:
999:
976:
956:
953:
948:
944:
930:
926:
922:
914:
910:
861:
853:
805:
802:
794:
790:
761:
757:
753:
745:
741:
706:
687:
681:
678:
675:
671:
667:
662:
659:
649:
645:
634:
633:
632:
616:
612:
604:
588:
566:
562:
547:
545:
540:
535:
531:
527:
522:
505:
502:
499:
475:
472:
461:
445:
437:
434:in which the
433:
428:
426:
422:
418:
414:
395:
392:
389:
386:
383:
380:
377:
374:
369:
365:
341:
333:
317:
303:
287:
283:
259:
252:
246:
243:
236:
228:
224:
210:
206:
202:
197:
193:
185:
184:
183:
166:
159:
153:
150:
144:
140:
126:
122:
118:
113:
109:
101:
100:
99:
98:
82:
75:
71:
67:
59:
56:equal 0) the
55:
37:
33:
23:
16:
2767:
2732:
2720:
2696:
2657:
2645:
2618:
2606:
2322:
2311:By checking
2310:
2121:
2117:
2098:
2092:
2085:
2067:
1895:
1846:uniform norm
1811:
1789:
1751:
1729:
1672:
1622:, such that
1565:
1559:
1548:
1357:
1355:
1344:
1167:
1111:
1023:
702:
553:
538:
523:
429:
309:
306:Introduction
274:
181:
69:
63:
15:
2114:Probability
1838:polynomials
1437:), so that
1366:and extend
66:mathematics
2805:Categories
2685:References
2080:See also:
2076:Variations
1844:under the
1802:See also:
1673:for every
727:such that
703:should be
2794:0072-5285
2759:0065-9282
2498:ν
2474:→
2462:ν
2393:→
2381:ν
2338:ν
2324:Corollary
2295:μ
2292:→
2283:μ
2256:μ
2227:μ
2207:∞
2204:→
2189:≥
2183:∀
2157:μ
2136:μ
2021:δ
1993:−
1982:∞
1967:∑
1951:ρ
1929:ρ
1914:μ
1820:μ
1738:μ
1684:∈
1658:μ
1648:∫
1633:φ
1578:μ
1529:≥
1488:∈
1477:for any
1462:≥
1450:φ
1374:φ
1272:≥
1260:φ
1203:μ
1124:∑
1071:∑
1032:φ
977:μ
957:μ
939:∞
934:∞
931:−
927:∫
904:Λ
884:Λ
862:∗
826:Λ
803:≥
784:Λ
735:Λ
715:Λ
589:μ
550:Existence
479:∞
446:μ
396:…
332:real line
318:μ
247:μ
219:∞
214:∞
211:−
207:∫
154:μ
135:∞
130:∞
127:−
123:∫
83:μ
54:cumulants
34:σ
2717:(1965).
2695:(1943).
2671:Archived
2556:See also
2356:satisfy
425:variance
1560:By the
542:is the
528:as the
436:support
97:moments
74:measure
2792:
2782:
2757:
2747:
2703:
539:μ
462:, for
334:, and
2674:(PDF)
2667:(PDF)
2598:Notes
2489:then
2274:then
1842:dense
776:and
2790:ISSN
2780:ISBN
2755:ISSN
2745:ISBN
2701:ISBN
1840:are
1806:and
421:mean
68:, a
2772:doi
2737:doi
2197:lim
2128:If
1900:as
1852:on
1564:, (
1168:If
1112:to
438:of
64:In
2807::
2788:.
2778:.
2753:.
2743:.
2669:.
2630:^
2319::
2110:.
2072:.
1727:.
1532:0.
1021:.
631:,
521:.
423:,
302:.
2796:.
2774::
2761:.
2739::
2709:.
2652:.
2640:.
2625:.
2539:)
2536:1
2533:,
2530:0
2527:(
2524:N
2502:n
2477:0
2471:]
2466:n
2458:[
2453:1
2450:+
2447:k
2444:2
2440:m
2435:;
2429:!
2426:k
2421:k
2417:2
2411:!
2408:)
2405:k
2402:2
2399:(
2390:]
2385:n
2377:[
2372:k
2369:2
2365:m
2342:n
2287:n
2262:,
2259:]
2253:[
2248:k
2244:m
2240:=
2236:]
2231:n
2223:[
2217:k
2213:m
2201:n
2192:0
2186:k
2161:n
2099:k
2093:k
2064:.
2050:n
2046:m
2042:)
2039:x
2036:(
2031:)
2028:n
2025:(
2014:!
2011:n
2004:n
2000:)
1996:1
1990:(
1977:0
1974:=
1971:n
1963:=
1960:)
1957:x
1954:(
1947:,
1944:x
1941:d
1938:)
1935:x
1932:(
1926:=
1923:)
1920:x
1917:(
1911:d
1872:]
1869:1
1866:,
1863:0
1860:[
1791:1
1776:]
1773:b
1770:,
1767:a
1764:[
1753:1
1715:)
1712:]
1709:b
1706:,
1703:a
1700:[
1697:(
1692:c
1688:C
1681:f
1655:d
1651:f
1645:=
1642:)
1639:f
1636:(
1610:]
1607:b
1604:,
1601:a
1598:[
1567:2
1553:)
1551:2
1549:(
1526:f
1522:,
1519:)
1516:]
1513:b
1510:,
1507:a
1504:[
1501:(
1496:c
1492:C
1485:f
1465:0
1459:)
1456:f
1453:(
1425:)
1422:]
1419:b
1416:,
1413:a
1410:[
1407:(
1402:c
1398:C
1359:1
1349:)
1347:1
1345:(
1339:.
1327:]
1324:b
1321:,
1318:a
1315:[
1295:P
1275:0
1269:)
1266:P
1263:(
1235:]
1232:b
1229:,
1226:a
1223:[
1181:k
1177:m
1153:.
1148:k
1144:m
1138:k
1134:a
1128:k
1095:k
1091:x
1085:k
1081:a
1075:k
1067:=
1064:)
1061:x
1058:(
1055:P
1009:]
1006:b
1003:,
1000:a
997:[
954:d
949:n
945:x
923:=
920:)
915:n
911:x
907:(
858:]
854:x
851:[
847:R
806:0
800:)
795:2
791:f
787:(
762:n
758:m
754:=
751:)
746:n
742:x
738:(
688:,
682:j
679:+
676:i
672:m
668:=
663:j
660:i
656:)
650:n
646:H
642:(
617:n
613:H
567:n
563:m
509:]
506:1
503:,
500:0
497:[
482:)
476:,
473:0
470:[
399:}
393:,
390:2
387:,
384:1
381:=
378:n
375::
370:n
366:x
362:{
342:M
288:n
284:M
260:.
256:)
253:x
250:(
244:d
240:)
237:x
234:(
229:n
225:M
203:=
198:n
194:m
167:.
163:)
160:x
157:(
151:d
145:n
141:x
119:=
114:n
110:m
38:2
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