Some Interesting Probability Limit Theorems by Shalosh B. Ekhad Theorem 1: Consider the Binomial distribution B(n,p) that is the probability distribution of the r.v. Number of Heads upon Tossing a loaded coin (Pr(H)=p) n times The even, 2r-th, moment, about the mean is (asymptotically, ignoring 1 ----, and beyond): 4 n / 2 2 2 r | (r - 1) (8 r p + 2 r - 8 r p + 2 p - 2 p - 1) r (-n p + n p) (2 r)! |1 - ------------------------------------------------- + \ 18 p (p - 1) n 3 4 3 3 3 3 2 3 r (r - 1) (r - 2) (320 r p + 20 r - 640 r p + 480 r p - 160 r p 2 2 4 2 3 2 2 2 - 60 r + 480 r p - 960 r p + 360 r p + 120 r p + 4 r p + 31 r 4 3 2 2 4 3 / + 28 r p - 56 r p + 24 r p - 12 p + 15 - 72 p - 84 p + 168 p ) / ( / 2 2 2 9720 p (p - 1) n ) - (r - 1) (r - 2) (r - 3) (3702 - 8483 r - 8676 p 2 6 4 5 2 + 7956 p - 71528 r p - 2100 r + 280 r + 23694 r p - 17634 r p 3 4 5 6 2 3 5 + 9888 p - 26064 p + 25344 p - 8448 p + 945 r + 4270 r + 214584 r p 3 3 2 3 3 3 4 2 - 8400 r p - 15960 r p + 7280 r p + 99960 r p - 15750 r p 2 2 2 3 2 4 3 4 + 17010 r p + 70560 r p - 217980 r p + 59408 r p - 208524 r p 2 6 3 6 3 5 4 6 2 5 - 73080 r p + 41440 r p - 124320 r p + 67200 r p + 219240 r p 4 5 4 4 4 3 4 2 4 - 201600 r p + 201600 r p - 67200 r p - 12600 r p + 12600 r p 5 6 5 5 5 4 5 3 5 2 + 17920 r p - 53760 r p + 67200 r p - 44800 r p + 16800 r p \ 5 / 3 3 3 | / r - 3360 r p) r / (18370800 p (p - 1) n )| / (r! 2 ) / / / 2 The variance (i.e. r=1 case) is, -n p + n p After dividing by it to the power r (i.e. normalizing it) we get that the normalized 2r-th moment is / 2 2 | (r - 1) (8 r p + 2 r - 8 r p + 2 p - 2 p - 1) r (2 r)! |1 - ------------------------------------------------- + r (r - 1) \ 18 p (p - 1) n 3 4 3 3 3 3 2 3 2 (r - 2) (320 r p + 20 r - 640 r p + 480 r p - 160 r p - 60 r 2 4 2 3 2 2 2 4 + 480 r p - 960 r p + 360 r p + 120 r p + 4 r p + 31 r + 28 r p 3 2 2 4 3 / 2 - 56 r p + 24 r p - 12 p + 15 - 72 p - 84 p + 168 p ) / (9720 p / 2 2 2 (p - 1) n ) - (r - 1) (r - 2) (r - 3) (3702 - 8483 r - 8676 p + 7956 p 6 4 5 2 3 - 71528 r p - 2100 r + 280 r + 23694 r p - 17634 r p + 9888 p 4 5 6 2 3 5 - 26064 p + 25344 p - 8448 p + 945 r + 4270 r + 214584 r p 3 3 2 3 3 3 4 2 - 8400 r p - 15960 r p + 7280 r p + 99960 r p - 15750 r p 2 2 2 3 2 4 3 4 + 17010 r p + 70560 r p - 217980 r p + 59408 r p - 208524 r p 2 6 3 6 3 5 4 6 2 5 - 73080 r p + 41440 r p - 124320 r p + 67200 r p + 219240 r p 4 5 4 4 4 3 4 2 4 - 201600 r p + 201600 r p - 67200 r p - 12600 r p + 12600 r p 5 6 5 5 5 4 5 3 5 2 + 17920 r p - 53760 r p + 67200 r p - 44800 r p + 16800 r p \ 5 / 3 3 3 | / r - 3360 r p) r / (18370800 p (p - 1) n )| / (r! 2 ) / / / The odd, (2r+1)-th, moment, about the mean is 2 r / r (2 r + 1) (-1 + 2 p) (-n p + n p) (2 r)! |- ---------------------- + (2 r + 1) (r - 1) (-1 + 2 p) \ 3 2 2 2 2 2 2 (40 r p + 70 r p + 24 p - 40 r p - 70 r p - 24 p - 3 + 10 r - 5 r) r/ (810 p (p - 1) n) - r (r - 1) (r - 2) (2 r + 1) (-1 + 2 p) (-54 + 147 r 2 4 2 3 4 2 - 36 p - 252 p + 28 r + 168 r p + 588 r p + 576 p - 288 p - 7 r 3 3 3 2 3 3 3 4 2 - 84 r - 168 r p + 2184 r p - 4032 r p + 2016 r p + 308 r p 2 2 2 3 2 4 3 4 4 4 + 2352 r p - 5320 r p + 2660 r p - 1512 r p + 756 r p + 448 r p 4 3 4 2 4 / 2 2 2 - 896 r p + 672 r p - 224 r p) / (204120 p (p - 1) n ) + (r - 1) / 2 (r - 2) (r - 3) (2 r + 1) (-1 + 2 p) (1242 + 1209 r + 216 p + 1944 p 6 4 5 6 2 3 - 64608 r p + 430 r - 300 r + 40 r + 1440 r p + 5580 r p + 12960 p 4 5 6 2 3 5 - 49680 p + 51840 p - 17280 p - 3728 r + 1395 r + 193824 r p 6 6 6 5 3 3 2 3 3 + 2560 r p - 7680 r p - 6030 r p - 8910 r p - 16560 r p 3 4 2 2 2 2 3 2 4 + 124380 r p + 834 r p - 5034 r p + 46328 r p - 117984 r p 3 4 6 2 6 3 6 4 + 50568 r p - 186804 r p + 2400 r p - 6400 r p + 9600 r p 6 2 6 3 6 3 5 4 6 - 480 r p - 37928 r p + 46440 r p - 139320 r p + 57760 r p 2 5 4 5 4 4 4 3 4 2 + 113784 r p - 173280 r p + 171240 r p - 53680 r p - 4440 r p 4 5 6 5 5 5 4 5 3 + 2400 r p + 21120 r p - 63360 r p + 69120 r p - 32640 r p 5 2 5 / 3 3 3 \ / r + 4680 r p + 1080 r p) r / (55112400 p (p - 1) n )| / (r! 2 ) / / / 2 The variance (again) is, -n p + n p After dividing by that variance to the power (r+1/2) (i.e. normalizing it) we get that the normalized 2r+1-th moment is / r (2 r + 1) (-1 + 2 p) (2 r)! |- ---------------------- + (2 r + 1) (r - 1) (-1 + 2 p) \ 3 2 2 2 2 2 2 (40 r p + 70 r p + 24 p - 40 r p - 70 r p - 24 p - 3 + 10 r - 5 r) r/ (810 p (p - 1) n) - r (r - 1) (r - 2) (2 r + 1) (-1 + 2 p) (-54 + 147 r 2 4 2 3 4 2 - 36 p - 252 p + 28 r + 168 r p + 588 r p + 576 p - 288 p - 7 r 3 3 3 2 3 3 3 4 2 - 84 r - 168 r p + 2184 r p - 4032 r p + 2016 r p + 308 r p 2 2 2 3 2 4 3 4 4 4 + 2352 r p - 5320 r p + 2660 r p - 1512 r p + 756 r p + 448 r p 4 3 4 2 4 / 2 2 2 - 896 r p + 672 r p - 224 r p) / (204120 p (p - 1) n ) + (r - 1) / 2 (r - 2) (r - 3) (2 r + 1) (-1 + 2 p) (1242 + 1209 r + 216 p + 1944 p 6 4 5 6 2 3 - 64608 r p + 430 r - 300 r + 40 r + 1440 r p + 5580 r p + 12960 p 4 5 6 2 3 5 - 49680 p + 51840 p - 17280 p - 3728 r + 1395 r + 193824 r p 6 6 6 5 3 3 2 3 3 + 2560 r p - 7680 r p - 6030 r p - 8910 r p - 16560 r p 3 4 2 2 2 2 3 2 4 + 124380 r p + 834 r p - 5034 r p + 46328 r p - 117984 r p 3 4 6 2 6 3 6 4 + 50568 r p - 186804 r p + 2400 r p - 6400 r p + 9600 r p 6 2 6 3 6 3 5 4 6 - 480 r p - 37928 r p + 46440 r p - 139320 r p + 57760 r p 2 5 4 5 4 4 4 3 4 2 + 113784 r p - 173280 r p + 171240 r p - 53680 r p - 4440 r p 4 5 6 5 5 5 4 5 3 + 2400 r p + 21120 r p - 63360 r p + 69120 r p - 32640 r p 5 2 5 / 3 3 3 \ / r + 4680 r p + 1080 r p) r / (55112400 p (p - 1) n )| / (r! 2 / / / 2 1/2 (-n p + n p) ) that obviously goes to 0 In particular, it follows that the normalized moments of the Binomial Distribution tend (2 r)! to the famous moments, ------, (for the even (2r)) and r r! 2 0, for the odd ones) of the Gaussian Distribution 2 x exp(- ----) 2 and hence is asymptotically normal. But we proved much more: we found not just the leading term 1 of the asymptotics, but its expansion up to term, ----, . 3 n Theorem 2: Consider the Mahonian Distribution, whose p.g.f. is n --------' ' | | i | | (1 - q ) | | | | i = 1 ------------------- n (1 - q) n! This p.g.f. describes both the prob. that a random permutation of size n will have a certain number of inversions, or a certain major index (in addition to some other more obscure permutation statistics). The average is, of course, 1/4 n(n - 1), .Centralizing we get that the 2r-th moment is / 3 2 r | 9 r (r - 1) (1/36 n + 1/8 n + 7/72 n) (2 r)! |1 - ----------- | 25 n \ 2 9 r (r - 1) (441 r - 205 r + 2321) + ----------------------------------- 2 61250 n 4 3 2 \ 3 (r - 1) (7938 r - 3132 r + 105048 r - 81702 r + 2550635) r| / - ---------------------------------------------------------------| / (r! 3 | / 3062500 n / r 2 ) After dividing by the variance to the power r 3 2 r (1/36 n + 1/8 n + 7/72 n) , (i.e. normalizing it) we get that the normalized 2r-th moment is / 2 | 9 r (r - 1) 9 r (r - 1) (441 r - 205 r + 2321) (2 r)! |1 - ----------- + ----------------------------------- | 25 n 2 \ 61250 n 4 3 2 \ 3 (r - 1) (7938 r - 3132 r + 105048 r - 81702 r + 2550635) r| / - ---------------------------------------------------------------| / (r! 3 | / 3062500 n / r 2 ) In particular, it follows that the Mahonian Distribution on permutations is asymptotically normal. A well-knwon fact, that can be found, e.g., in : Feller v. I, 3rd ed., p. 257 But he only talked abiut the leading term, while we have established the asymptotic to, 4, terms . Of course, by symmetry, all the odd moments are zero but we can derive this directly, getting that indeed the odd moments eqaul 0 Theorem 3: Consider the q-Catalan Distribution whose p.g.f. is / n \ | --------' (n + i)| |' | | 1 - q | (1 - q) n! (n + 1)! | | | ------------| | | | i | | | | 1 - q | \ i = 1 / --------------------------------------------- (n + 1) (1 - q ) (2 n)! This p.g.f. describes both the distribution of so called Catalan words according to the major index. statistics). The average is, of course, 1/2 n(n - 1), .Centralizing we get that the 2r-th moment is / 3 2 r | 3 r (r - 1) (1/6 n + 7/12 n + 2/3 n) (2 r)! |1 - ----------- | 10 n \ 2 r (r - 1) (63 r - 75 r + 248) + ------------------------------ 2 1400 n 4 3 2 \ (r - 1) (189 r - 486 r + 2229 r - 1396 r + 2090) r| / r - -----------------------------------------------------| / (r! 2 ) 3 | / 42000 n / 3 2 The variance is, 1/6 n + 7/12 n + 2/3 n After dividing by the variance to the power r (i.e. normalizing it) we get that the normalized 2r-th moment is / 2 | 3 r (r - 1) r (r - 1) (63 r - 75 r + 248) (2 r)! |1 - ----------- + ------------------------------ | 10 n 2 \ 1400 n 4 3 2 \ (r - 1) (189 r - 486 r + 2229 r - 1396 r + 2090) r| / r - -----------------------------------------------------| / (r! 2 ) 3 | / 42000 n / In particular, it follows that the q-Catalan Distribution is asymptotically normal. As recently proved by W.Y.C. Chen, C.J. Wang, and L. X.W. Wang in their article: "The Limiting Distribution of the Coefficients of the q-Catalan Numbers . But their result only implies the leading term, of the asymptotics and we have established the asymptotic to, 4, terms . Of course, by symmetry, all the odd moments are zero but we can derive this directly, getting that indeed all the odd moments eqaul 0 This took, 46.919, seconds