Explicit [rigorously-proved] (polynomial) Formuals for all the Mixed Moments\ up to the (, 5, 5, 5, ) of the Sucker-Bets Distribution with 3 Decks, each with n cards, where all \ the 3n cards have distinct numbers and a proof that the scaled mixed moments converge, as n goes to infinity, t\ o the corresponding mixed moments of the Ekhad-Zeilberger trivariate continuous distriution, i.e. the limit, as c goe\ s to 1 from below of the distribution whose probability density function is 2 2 2 x y z 1/2 2 1/2 1/4 exp(- ---- - ---- - ---- - (x y + x z + y z) c) 2 (-c + 1) (1 + 2 c) 2 2 2 / 3/2 / (Pi (c + 1)) / By Shalosh B. Ekhad (3 n)! Consider the set of all, ------, 3 (n!) words in the alphabet {1,2,3} with n 1's, n 2's and n 3's note that from each such word we can form a set of three card-decks, each wi\ th n cards,with denominations from 1 to 3n by forming the first deck with cards correpsonding to the locations of 1 the second deck with cards correpsonding to the locations of 2 the third deck with cards correpsonding to the locations of 3 and define three statistics, S12, S23, S31, where S12(w)= (Number of occurrences of 1 before 2) Minus (Number of occurrences of\ 2 before 1) S23(w)= (Number of occurrences of 2 before 3) Minus (Number of occurrences of\ 3 before 2) S31(w)= (Number of occurrences of 3 before 1) Minus (Number of occurrences of\ 1 before 3) Note that "before" does not (necessarily) means "right before" Note that the triples of decks of cards corresponding to words for which all\ of S12(w),S23(w), S31(w) are strictly positive are sucker's bets The following are explicit expressions for all the mixed moments M(i,j,k), f\ or 0<=i<=j<=k<=5, and i+j+k even Note that it is always 0 when i+j+k is odd, and by symmetry you can get all \ of them. The variance of each of S12, S23, S31 is 2 n (2 n + 1) ------------ 3 The fourth-moment of each of S12, S23, S31 is 3 2 n (2 n + 1) (10 n - n - 4) ---------------------------- 15 Hence the kurtosis is 2 3 (10 n - n - 4) ----------------- 5 n (2 n + 1) whose limit, as n goes to infinity is 3 as it should!, since they are each asympotically normal. However they are far from independent! The covariance between any two of them is 3 n - ---- 3 Hence the correlation coefficient between any two of these random variables \ is n - ------- 2 n + 1 whose limit is -1/2 Let M(i,j,k) be the (i,j,k) mixed moment for the statistics S12(w), S23(w), \ and S31(w) M(0, 0, 0) = 1 2 n (2 n + 1) M(0, 0, 2) = ------------ 3 3 2 n (2 n + 1) (10 n - n - 4) M(0, 0, 4) = ---------------------------- 15 3 n M(0, 1, 1) = - ---- 3 4 2 n (10 n - n - 4) M(0, 1, 3) = - ------------------ 15 4 5 4 3 2 n (140 n - 112 n - 115 n + 92 n + 32 n - 16) M(0, 1, 5) = - ------------------------------------------------- 63 3 3 2 n (30 n + 14 n + 3 n - 2) M(0, 2, 2) = ---------------------------- 45 3 6 5 4 3 2 n (560 n - 112 n - 146 n - 3 n - 16 n + 40 n - 8) M(0, 2, 4) = ------------------------------------------------------- 315 3 6 5 4 3 2 n (490 n - 210 n - 357 n + 42 n + 144 n - 12 n + 8) M(0, 3, 3) = - --------------------------------------------------------- 315 3 9 8 7 6 5 4 M(0, 3, 5) = - n (5600 n - 8120 n - 2406 n + 6237 n + 3060 n - 3336 n 3 2 - 1664 n + 1280 n - 432 n + 96)/945 3 9 8 7 6 5 4 M(0, 4, 4) = n (26600 n - 31920 n - 2568 n + 26472 n - 7899 n - 6480 n 3 2 + 3928 n - 5232 n + 2304 n - 480)/4725 3 12 11 10 9 8 M(0, 5, 5) = - n (157080 n - 437360 n + 261624 n + 333608 n - 301281 n 7 6 5 4 3 2 - 162960 n - 47736 n + 254240 n + 44928 n - 183584 n + 119808 n - 44352 n + 8064)/6237 3 2 n (11 n + 2 n + 2) M(1, 1, 2) = - -------------------- 45 3 6 5 4 3 2 n (140 n - 420 n - 13 n + 144 n + 12 n + 40 n - 8) M(1, 1, 4) = -------------------------------------------------------- 315 3 6 5 4 3 2 n (70 n + 28 n + 45 n - 52 n + 34 n - 28 n + 8) M(1, 2, 3) = - ----------------------------------------------------- 315 3 9 8 7 6 5 4 M(1, 2, 5) = - n (1400 n - 2800 n + 2588 n - 861 n + 226 n - 102 n 3 2 - 760 n + 1024 n - 496 n + 96)/945 3 9 8 7 6 5 4 M(1, 3, 4) = n (1400 n - 7560 n - 1350 n + 6507 n + 2472 n - 4908 n 3 2 + 4768 n - 4680 n + 2256 n - 480)/4725 3 12 11 10 9 8 M(1, 4, 5) = - n (46200 n - 141680 n + 330352 n - 151580 n - 423201 n 7 6 5 4 3 2 + 479004 n - 127692 n - 171016 n + 574328 n - 774688 n + 552960 n - 222912 n + 40320)/31185 3 6 5 4 3 2 n (140 n + 294 n - 72 n - 9 n - 54 n + 24 n - 8) M(2, 2, 2) = ------------------------------------------------------ 315 3 9 8 7 6 5 4 M(2, 2, 4) = n (5600 n + 8120 n - 8686 n - 5874 n + 6475 n - 3390 n 3 2 + 5216 n - 4496 n + 2240 n - 480)/4725 3 9 8 7 6 5 4 M(2, 3, 3) = - n (2800 n + 3290 n - 9918 n + 5757 n - 3516 n + 6594 n 3 2 - 5776 n + 4072 n - 2208 n + 480)/4725 3 12 11 10 9 8 M(2, 3, 5) = - n (92400 n - 33880 n - 435402 n + 626956 n - 90681 n 7 6 5 4 3 2 - 204888 n + 191454 n - 465956 n + 727464 n - 744976 n + 529920 n - 222336 n + 40320)/31185 3 12 11 10 9 8 M(2, 4, 4) = n (462000 n + 773080 n - 2399408 n + 1100176 n + 447582 n 7 6 5 4 3 + 1099791 n - 2867640 n + 3347384 n - 3910048 n + 3818240 n 2 - 2599968 n + 1086336 n - 201600)/155925 3 15 14 13 12 M(2, 5, 5) = - n (33633600 n - 12612600 n - 364283920 n + 823446624 n 11 10 9 8 - 371158944 n - 572235067 n + 516766182 n + 110459702 n 7 6 5 4 - 578801744 n + 1443895728 n - 2642784288 n + 3369542272 n 3 2 - 3039561984 n + 1848398464 n - 675202560 n + 111444480)/2837835 3 11 10 9 8 7 M(3, 3, 4) = - n (141680 n + 80014 n - 716342 n + 899763 n - 842358 n 6 5 4 3 2 + 1120788 n - 1397660 n + 1433408 n - 1258768 n + 849024 n - 359424 n + 67200)/51975 3 15 14 13 12 M(3, 4, 5) = - n (42042000 n + 29429400 n - 288648360 n + 454726272 n 11 10 9 8 - 616876650 n + 1522875611 n - 3466829696 n + 6088185732 n 7 6 5 4 - 9388493136 n + 13388708088 n - 16915309920 n + 17936178880 n 3 2 - 15034561408 n + 9035218560 n - 3339138048 n + 557222400)/14189175 3 15 14 13 M(4, 4, 4) = n (140140000 n + 504504000 n - 1499618120 n 12 11 10 9 - 1096791696 n + 7342193508 n - 10641870640 n + 11263147417 n 8 7 6 5 - 15566404476 n + 22207220072 n - 27552084528 n + 31164782784 n 4 3 2 - 30989982368 n + 25030908160 n - 14872399872 n + 5518608384 n - 928704000)/23648625 3 18 17 16 15 M(4, 5, 5) = - n (39239200 n + 66146080 n - 816055240 n + 1114633520 n 14 13 12 11 + 3208398492 n - 13589761044 n + 25028291837 n - 38043392560 n 10 9 8 7 + 62580129596 n - 103184180072 n + 157753326632 n - 224678523360 n 6 5 4 3 + 293133737664 n - 336053442624 n + 322828696448 n - 243844376832 n 2 + 132045454336 n - 44452356096 n + 6864979968)/2837835 It follows that the scaled mixed moments, M(i,j,k)/M(2,0,0)^((i+j+k)/2)), as\ n goes to infinity, let's call these limits S(i,j,k), are as follows. S(0, 0, 0) = 1 S(0, 0, 2) = 1 S(0, 0, 4) = 3 S(0, 1, 1) = -1/2 S(0, 1, 3) = -3/2 S(0, 1, 5) = -15/2 S(0, 2, 2) = 3/2 S(0, 2, 4) = 6 S(0, 3, 3) = -21/4 S(0, 3, 5) = -30 S(0, 4, 4) = 57/2 S(0, 5, 5) = -765/4 S(1, 1, 2) = 0 S(1, 1, 4) = 3/2 S(1, 2, 3) = -3/4 S(1, 2, 5) = -15/2 S(1, 3, 4) = 3/2 S(1, 4, 5) = -45/4 S(2, 2, 2) = 3/2 S(2, 2, 4) = 6 S(2, 3, 3) = -3 S(2, 3, 5) = -45/2 S(2, 4, 4) = 45/2 S(2, 5, 5) = -135 S(3, 3, 4) = 0 S(3, 4, 5) = -135/4 S(4, 4, 4) = 135/2 S(4, 5, 5) = -945/4 Consider the trivariate distribution whose probability density function in x\ ,y,z, is 2 2 2 x y z 1/2 2 1/2 1/4 exp(- ---- - ---- - ---- - (x y + x z + y z) c) 2 (-c + 1) (1 + 2 c) 2 2 2 / 3/2 / (Pi (c + 1)) / where c is between 0 and 1. We claim that the limit as c goes to 1, from below of the above tri-variate \ continuous distribution is the asymptotic limit of the (scaled) Sucker-Bets tri-variate statistics \ as n goes to infinity Let Mc(a1,a2,a3) be its scaled (i,j,k)-mixed moment (i.e. the (a1,a2,a3) mix\ ed moment, divided by the standard deviation to the power a1+a2+a3 Using the amazing Apagodu-Zeilberger multi-variate Alkvist-Zeilberger algori\ thm the following theorem was discovered and proved (but we omit the proof) Theorem: If a1+a2 is even we have the following recurrence 3 3 3 2 2 3 Mc(a1, a2, 2 a3) = 2 (a1 c + a2 c - 2 a3 c + a1 c + a2 c + 3 c + 8 a3 c 2 / 2 + c + 4 a3 - 10 c - 5) Mc(a1, a2, -2 + 2 a3) / (c + 1) - (1 + 2 c) ( / 2 4 4 2 4 2 4 3 3 a1 c - 2 a1 a2 c + a2 c - 4 a3 c + 8 a1 a3 c + 8 a2 a3 c 2 3 4 2 3 2 3 - 24 a3 c + 12 a3 c + 8 a1 a3 c - 14 a1 c + 8 a2 a3 c - 14 a2 c 3 4 2 2 2 2 3 + 88 a3 c - 9 c - 14 a1 c - 14 a2 c + 48 a3 c + 8 a3 c - 82 c 2 2 + 24 a3 - 168 a3 c - 14 c - 84 a3 + 150 c + 75) Mc(a1, a2, 2 a3 - 4) / 4 2 / (c + 1) + 4 (1 + 2 c) (a3 - 2) (2 a3 - 5) ( / 3 2 2 2 3 2 -2 a3 c + a1 c + a2 c - 4 a3 c + 4 c + 4 a3 c + 10 c + 4 a3 - 9 c - 9 / 5 3 ) Mc(a1, a2, 2 a3 - 6) / (c + 1) + 4 (-1 + c) (1 + 2 c) (2 a3 - 5) / / 5 (2 a3 - 7) (a3 - 2) (a3 - 3) Mc(a1, a2, 2 a3 - 8) / (c + 1) / If a1+a2 is odd we have the following recurrence 3 3 3 2 2 3 Mc(a1, a2, 2 a3 + 1) = 2 (a1 c + a2 c - 2 a3 c + a1 c + a2 c + 2 c 2 / 2 + 8 a3 c + c + 4 a3 - 6 c - 3) Mc(a1, a2, -1 + 2 a3) / (c + 1) - / 2 4 4 2 4 2 4 3 (1 + 2 c) (a1 c - 2 a1 a2 c + a2 c - 4 a3 c + 8 a1 a3 c 3 2 3 4 2 3 2 + 8 a2 a3 c - 24 a3 c + 8 a3 c + 8 a1 a3 c - 10 a1 c + 8 a2 a3 c 3 3 4 2 2 2 2 - 10 a2 c + 64 a3 c - 4 c - 10 a1 c - 10 a2 c + 48 a3 c + 8 a3 c 3 2 2 - 44 c + 24 a3 - 120 a3 c - 10 c - 60 a3 + 78 c + 39) / 4 2 Mc(a1, a2, 2 a3 - 3) / (c + 1) + 4 (1 + 2 c) (2 a3 - 3) (a3 - 2) ( / 3 2 2 2 3 2 -2 a3 c + a1 c + a2 c - 4 a3 c + 3 c + 4 a3 c + 8 c + 4 a3 - 7 c - 7) / 5 3 Mc(a1, a2, 2 a3 - 5) / (c + 1) + 4 (-1 + c) (1 + 2 c) (2 a3 - 3) / / 5 (2 a3 - 5) (a3 - 2) (a3 - 3) Mc(a1, a2, 2 a3 - 7) / (c + 1) / Note that this enables a very fast computation of these mixed momnets Taking the limits as c goes to 1 (from below),let's call S(a1,a2,a3)=Mc(a1\ ,a2,a3,1), we get the following corollary Corollary: If a1+a2 is even we have the following recurrence 2 S(a1, a2, 2 a3) = (a1 + a2 + 5 a3 - 11/2) S(a1, a2, -2 + 2 a3) + (-3/16 a1 2 2 + 3/8 a1 a2 - 3 a1 a3 - 3/16 a2 - 3 a2 a3 - 33/4 a3 + 21/4 a1 + 21/4 a2 + 27 a3 - 45/2) S(a1, a2, 2 a3 - 4) + 9/8 (a3 - 2) (2 a3 - 5) (a1 + a2 + 2 a3 - 4) S(a1, a2, 2 a3 - 6) If a1+a2 is odd we have the following recurrence 2 S(a1, a2, 2 a3 + 1) = (a1 + a2 + 5 a3 - 11/2) S(a1, a2, -1 + 2 a3) + (-3/16 a1 2 2 + 3/8 a1 a2 - 3 a1 a3 - 3/16 a2 - 3 a2 a3 - 33/4 a3 + 21/4 a1 + 21/4 a2 + 27 a3 - 45/2) S(a1, a2, 2 a3 - 3) + 9/8 (a3 - 2) (2 a3 - 5) (a1 + a2 + 2 a3 - 4) S(a1, a2, 2 a3 - 5) This enables very fast computation of as many mixed moments as we wish. In particular it confirms our conjecture that up to all mixed moments (i,j,k\ ) with, i,j,k <=, 5 the limiting distribution of the 3-Deck Sucker Bets statistics (each with n \ cards) tends to this one The mixed moments, even for c=1, do not have closed form, but the diagonal, \ S(2n,2n,2n) surprisingly do! The following theorem also follows from the Apagodu-Zeilberger multi-variate\ Alkvist-Zeilberger algorithm Theorem: for each non-negative integer, we have (3 n)! (2 n)! S(2 n, 2 n, 2 n) = ------------- n 2 8 (n!) This ends this article.