A generalization of the Central Limit Theorem by Shalosh B. Ekhad Consider the centralized distribution obtained by summing n identically-distributed independent (abbreviated :i.i.d.) r.v. with finite moments, M[1]=0, M[2],M[3], ... The 2r-th moment, divided by the (2r)-th moment of of the Standard Normal Distribution, (2r)!/(r!2^r) divided by M[2]^r has 1 has asymptotics up to order, ---- 4 n as follows. 2 3 2 r (-1 + r) (2 r M[3] + 3 M[2] M[4] - 9 M[2] - 4 M[3] ) 1 + 1/18 -------------------------------------------------------- + 1/9720 r 3 M[2] n 3 4 4 2 4 (-1 + r) (r - 2) (20 r M[3] + 940 r M[3] - 240 r M[3] 2 2 2 2 2 - 405 M[2] M[4] + 2160 M[2] M[4] M[3] + 180 r M[2] M[4] M[3] 2 2 4 + 216 M[2] r M[3] M[5] - 1260 M[2] r M[4] M[3] - 1200 M[3] 2 3 2 4 2 2 - 540 r M[2] M[3] - 810 M[2] r M[4] + 135 M[2] r M[4] 3 2 2 4 + 1620 r M[2] M[3] - 648 M[2] M[3] M[5] + 810 M[2] M[4] 3 2 3 6 6 / 6 - 1080 M[2] M[3] + 108 M[2] M[6] + 1215 r M[2] - 405 M[2] ) / (M[2] / 2 2 3 n ) + 1/18370800 (r - 3) (r - 2) (-1 + r) r (68040 r M[2] M[3] M[4] M[5] 3 4 2 - 612360 M[2] r M[3] M[4] M[5] + 4082400 M[2] M[3] M[4] 6 2 7 5 + 1530900 r M[2] M[3] - 893025 M[2] r M[4] - 1224720 M[2] M[3] M[5] 4 4 5 2 2 - 136080 M[2] M[4] M[6] + 5292000 M[2] M[4] M[3] - 127575 M[2] r M[4] 6 2 2 7 2 3 4 4 - 1105650 M[2] r M[3] + 382725 M[2] r M[4] - 18900 M[2] r M[3] 6 4 2 2 2 - 1881600 M[3] - 77760 M[2] M[3] M[7] - 3402000 M[2] M[4] M[3] 2 3 3 2 - 1814400 M[2] M[5] M[3] + 453600 M[2] M[3] M[6] 2 3 3 2 3 4 + 14175 r M[2] M[4] - 1341900 r M[2] M[3] 2 2 2 4 2 + 2097900 M[2] r M[4] M[3] - 4422600 M[2] r M[3] M[4] 4 4 4 2 5 2 + 6300 r M[2] M[4] M[3] + 20412 M[2] r M[5] + 637875 M[2] r M[4] 6 3 - 102060 r M[2] M[6] + 1360800 M[2] M[3] M[4] M[5] 2 2 2 2 2 3 2 - 425250 r M[2] M[4] M[3] + 22680 r M[2] M[3] M[6] 3 4 3 3 5 2 + 2910600 M[2] r M[3] - 127575 M[2] r M[4] - 204120 M[2] r M[3] M[5] 6 3 2 2 2 3 + 255150 M[2] r M[3] - 226800 r M[2] M[5] M[3] 3 3 4 4 2 2 + 264600 r M[2] M[3] + 1530900 M[2] r M[3] M[4] 2 4 3 2 + 1127700 r M[2] M[4] M[3] - 204120 M[2] r M[3] M[6] 4 4 + 19440 M[2] r M[3] M[7] + 34020 M[2] r M[4] M[6] 3 2 2 2 5 + 28350 r M[2] M[4] M[3] + 1020600 M[2] r M[3] M[5] 2 3 4 3 2 + 1118880 M[2] r M[5] M[3] - 170100 M[2] r M[3] M[4] 4 3 2 3 - 4019400 r M[2] M[4] M[3] + 15120 r M[2] M[5] M[3] 3 4 3 6 6 - 138600 r M[2] M[4] M[3] + 99400 r M[3] + 1664320 r M[3] 2 6 4 6 3 4 - 579600 r M[3] - 8400 r M[3] - 2268000 M[2] M[3] 6 2 6 7 - 680400 M[2] M[3] + 204120 M[2] M[6] + 510300 M[2] M[4] 5 2 3 3 5 6 5 - 765450 M[2] M[4] + 283500 M[2] M[4] + 280 r M[3] + 7290 M[2] M[8] 4 2 9 2 9 / 9 3 - 81648 M[2] M[5] + 382725 r M[2] - 382725 r M[2] ) / (M[2] n ) + / 3 2 3 1/1322697600 (45541440 M[2] r M[4] M[5] M[3] 5 6 2 - 2566080 M[2] r M[3] M[4] M[7] - 38374560 M[2] M[6] M[3] r 6 2 2 2 2 4 + 9797760 M[2] M[6] M[3] r - 571536000 M[2] M[4] M[3] 4 2 2 4 2 - 4898880 M[2] r M[4] M[6] M[3] + 29121120 M[2] r M[4] M[6] M[3] 4 3 2 3 3 3 + 272160 M[2] r M[4] M[6] M[3] - 4717440 M[2] r M[4] M[5] M[3] 3 4 3 6 4 + 181440 M[2] r M[4] M[5] M[3] + 163069200 r M[2] M[3] 5 3 4 3 6 - 544320 M[2] M[3] M[5] r + 247484160 M[2] r M[3] 5 3 5 3 3 - 195955200 M[2] M[3] M[5] + 9797760 M[2] M[3] M[5] r 5 3 5 3 2 + 186157440 M[2] M[3] M[5] r - 64774080 M[2] M[3] M[5] r 7 2 2 7 3 2 + 143904600 M[2] r M[4] M[3] - 34700400 M[2] r M[4] M[3] 7 4 2 8 2 2 + 3061800 M[2] r M[4] M[3] - 16839900 M[2] M[4] r 8 2 3 8 2 + 2296350 M[2] M[4] r + 40568850 M[2] M[4] r 5 2 5 + 326592 M[2] M[3] M[5] M[6] r - 3592512 M[2] M[3] M[5] M[6] r 4 2 8 - 85730400 M[2] M[5] M[4] M[3] - 133056000 M[3] 5 2 2 4 6 - 1020600 M[2] M[4] M[3] r + 87480 M[2] M[4] M[8] r 5 2 2 3 5 2 2 2 + 18370800 M[2] M[4] M[3] r - 121451400 M[2] M[4] M[3] r 5 2 2 6 12 + 349045200 M[2] M[4] M[3] r - 437400 M[2] M[4] M[8] + 459270 M[2] 4 2 2 4 2 - 7348320 M[2] M[5] M[4] M[3] r + 43681680 M[2] M[5] M[4] M[3] r 4 2 3 7 + 408240 M[2] M[5] M[4] M[3] r + 8573040 M[2] M[4] M[6] r 7 2 3 4 6 - 1224720 M[2] M[4] M[6] r - 4485600 M[2] r M[3] 3 2 6 3 3 6 - 122512320 M[2] r M[3] + 31600800 M[2] r M[3] 3 5 6 6 + 332640 M[2] r M[3] - 115123680 r M[2] M[3] M[4] M[5] 3 6 6 9 5 2 - 10080 M[2] r M[3] - 7960680 r M[2] M[6] - 2694384 M[2] M[4] M[5] r 5 2 2 3 6 + 244944 M[2] M[4] M[5] r - 2449440 r M[2] M[3] M[4] M[5] 2 6 6 3 + 29393280 r M[2] M[3] M[4] M[5] + 30618000 M[2] M[4] 7 2 12 + 4898880 M[2] M[3] M[7] r - 6889050 r M[2] 7 2 7 - 699840 M[2] M[3] M[7] r - 8398080 M[2] M[3] M[7] 6 5 4 5 2 2 + 340200 M[2] r M[3] + 204120 M[2] r M[6] M[4] 5 2 6 + 7348320 M[2] M[4] M[5] + 508032000 M[2] M[4] M[3] 6 6 3 - 429609600 M[2] M[4] M[3] r - 27468000 M[2] M[4] M[3] r 6 2 6 5 + 149600640 M[2] M[4] M[3] r - 151200 M[2] M[4] M[3] r 6 4 6 6 + 2805600 M[2] M[4] M[3] r + 3360 M[2] M[4] M[3] r 3 3 2 4 3 3 2 + 113400 M[2] M[4] M[3] r + 190512000 M[2] M[4] M[3] 3 3 2 2 3 3 2 + 28463400 M[2] M[4] M[3] r - 120884400 M[2] M[4] M[3] r 3 3 2 3 4 2 - 2948400 M[2] M[4] M[3] r - 57153600 M[2] M[4] M[6] M[3] 5 2 2 9 3 2 - 367416000 M[2] M[4] M[3] + 18370800 M[2] r M[3] 9 4 2 7 4 3 - 3061800 M[2] r M[3] + 11664 M[2] M[10] - 10886400 M[2] M[7] M[3] 6 3 4 3 - 23984100 r M[2] M[4] + 5546880 M[2] M[7] M[3] r 4 3 2 9 2 - 933120 M[2] M[7] M[3] r - 12247200 M[2] M[3] 3 6 3 2 6 3 - 510300 r M[2] M[4] + 6123600 r M[2] M[4] 4 3 3 2 5 + 51840 M[2] M[7] M[3] r - 182891520 M[2] M[5] M[3] 6 2 6 4 4 6 3 4 + 81648 r M[2] M[6] - 5556600 M[2] r M[3] + 35267400 M[2] r M[3] 9 6 2 4 + 8573040 M[2] M[6] - 108750600 M[2] r M[3] 6 2 7 + 48988800 M[2] M[6] M[3] - 14696640 M[2] M[4] M[6] 6 2 3 2 2 4 - 816480 M[2] M[6] M[3] r + 426157200 M[2] r M[4] M[3] 6 6 + 38880 M[2] M[9] M[3] r - 194400 M[2] M[9] M[3] 10 2 2 2 2 4 + 18370800 M[2] r M[4] - 125685000 M[2] r M[4] M[3] 2 5 2 4 2 4 2 4 + 37800 M[2] r M[4] M[3] - 1323000 M[2] r M[4] M[3] 2 3 2 4 10 + 18333000 M[2] r M[4] M[3] + 9185400 M[2] M[4] 10 3 5 - 4592700 M[2] r M[4] + 6998400 M[2] M[3] M[4] M[7] 7 2 8 3 + 5143824 M[2] r M[5] + 3674160 M[2] M[3] M[5] r 8 2 8 - 26943840 M[2] M[3] M[5] r + 64910160 M[2] M[3] M[5] r 2 9 2 7 2 3 6 - 39803400 r M[2] M[3] - 8817984 M[2] M[5] - 203212800 M[2] M[3] 12 5 2 + 1148175 r M[2] + 233280 M[2] r M[3] M[4] M[7] 10 3 12 - 22963500 M[2] r M[4] + 3444525 r M[2] 7 2 7 2 - 259232400 M[2] M[4] M[3] r + 171460800 M[2] M[4] M[3] 4 2 2 8 - 34292160 M[2] M[5] M[3] + 787320 M[2] M[8] 4 2 2 6 + 17472672 M[2] M[5] M[3] r + 146966400 M[2] M[3] M[4] M[5] 4 2 2 2 4 2 2 3 - 2939328 M[2] M[5] M[3] r + 163296 M[2] M[5] M[3] r 4 2 4 4 3 4 4 4 - 765450 M[2] r M[4] + 42525 M[2] r M[4] + 4550175 M[2] r M[4] 8 2 5 5 - 51438240 M[2] M[3] M[5] + 12096 M[2] r M[5] M[3] 2 4 5 2 3 5 - 423360 M[2] r M[5] M[3] + 5866560 M[2] r M[5] M[3] 2 5 2 2 5 + 136370304 M[2] r M[5] M[3] - 40219200 M[2] r M[5] M[3] 8 2 8 7 8 2 - 32148900 M[2] M[4] + 80 M[3] r - 49409920 M[3] r 8 5 2 + 124612800 M[3] r - 2245320 M[2] M[6] M[4] r 5 2 5 2 + 6123600 M[2] M[6] M[4] - 641520 M[2] M[3] M[8] r 5 2 5 2 2 8 6 + 1749600 M[2] M[3] M[8] + 58320 M[2] M[3] M[8] r - 4480 M[3] r 8 5 3 3 8 4 + 106400 M[3] r + 304819200 M[2] M[4] M[5] M[3] - 1388800 M[3] r 4 4 8 8 3 - 8930250 M[2] M[4] - 262440 M[2] r M[8] + 10755920 M[3] r 5 6 4 + 9797760 M[2] M[3] M[5] M[6] - 95256000 M[2] M[3] 4 5 4 2 9 - 226800 M[2] r M[4] M[3] + 1837080 r M[2] M[6] 4 4 4 4 - 624607200 M[2] M[4] M[3] r + 571536000 M[2] M[4] M[3] 7 2 2 4 4 4 - 734832 M[2] r M[5] + 5670000 M[2] M[4] M[3] r 4 4 3 4 4 2 - 55566000 M[2] M[4] M[3] r + 266490000 M[2] M[4] M[3] r 3 4 3 4 - 32235840 M[2] M[6] M[3] r + 50803200 M[2] M[6] M[3] 3 4 2 6 2 + 7590240 M[2] M[6] M[3] r - 408240 M[2] M[6] 3 4 4 3 4 3 + 30240 M[2] M[6] M[3] r - 786240 M[2] M[6] M[3] r 9 2 6 + 36741600 r M[2] M[3] - 699840 M[2] M[5] M[7] 6 3 3 + 139968 M[2] M[5] M[7] r - 193415040 M[2] r M[4] M[5] M[3] ) (r - 4) / 12 4 (r - 3) (r - 2) (-1 + r) r / (M[2] n ) / If that distribution is symmetric, and the second moment is normalized to 1, then it equals r (-1 + r) (-9 + 3 M[4]) 1 + 1/18 ------------------------ + 1/9720 r (-1 + r) (r - 2) ( n 2 2 -405 - 405 M[4] - 810 M[4] r + 135 r M[4] + 810 M[4] + 108 M[6] + 1215 r) / 2 2 / n + 1/18370800 (r - 3) (r - 2) (-1 + r) r (637875 r M[4] / 2 2 - 102060 r M[6] + 382725 r - 765450 M[4] - 136080 M[4] M[6] - 382725 r 3 2 3 + 14175 M[4] r + 510300 M[4] + 204120 M[6] + 283500 M[4] + 7290 M[8] 2 2 2 3 - 893025 M[4] r - 127575 M[4] r + 382725 M[4] r - 127575 r M[4] / 3 2 + 34020 r M[4] M[6]) / n + 1/1322697600 (459270 + 40568850 r M[4] / 2 + 1837080 r M[6] - 7960680 r M[6] - 262440 r M[8] + 1148175 r 2 3 2 - 32148900 M[4] - 14696640 M[4] M[6] + 3444525 r - 6889050 r 3 2 3 3 + 6123600 M[4] r - 510300 M[4] r - 437400 M[4] M[8] + 9185400 M[4] 4 2 4 3 3 - 765450 M[4] r + 42525 M[4] r + 8573040 M[6] + 30618000 M[4] 2 4 + 787320 M[8] - 408240 M[6] - 8930250 M[4] + 11664 M[10] 2 2 2 3 - 22963500 M[4] r - 16839900 M[4] r + 2296350 M[4] r 2 3 2 4 + 18370800 M[4] r - 4592700 M[4] r + 81648 r M[6] + 4550175 r M[4] 2 3 + 6123600 M[6] M[4] - 23984100 r M[4] + 87480 M[4] M[8] r 2 2 - 1224720 M[4] M[6] r - 2245320 M[6] M[4] r + 8573040 r M[4] M[6] 2 2 / 4 + 204120 r M[6] M[4] ) (r - 4) (r - 3) (r - 2) (-1 + r) r / n / if we want the, 1/n, term to vanish we need the, 4, -th moment to be, 3 1 if we also want the, ----, term to vanish 2 n we need the, 6, -th moment to be, 15 1 if we also want the, ----, term to vanish 3 n we need the, 8, -th moment to be, 105 1 if we also want the, ----, term to vanish 4 n we need the, 10, -th moment to be, 945 This took, 813.150, seconds of CPU time