EXCHANGEABLE PAIRS OF BERNOULLI RANDOM VARIABLES, KRAWTCHOUCK POLYNOMIALS, AND EHRENFEST URNS

This paper derives characterizations of bivariate binomial distributions of the Lancaster form with Krawtchouk polynomial eigenfunctions. These have been characterized by Eagleson, and we give two further characterizations with a more probabilistic flavour: the first as sums of correlated Bernoulli variables; and the second as the joint distribution of the number of balls of one colour at consecutive time points in a generalized Ehrenfest urn. We give a self‐contained development of Krawtchouck polynomials and Eagleson’s theorem.

[1]  Review of ``Classical and quantum orthogonal polynomials in one variable. "by Mourad E.H. Ismail , 2011 .

[2]  H. Mizukawa.  Finite Gelfand Pair Approaches for Ehrenfest Diffusion Model , 2010, 1009.1205.

[3]  Bob Griffiths,et al.  Stochastic processes with orthogonal polynomial eigenfunctions , 2009, J. Comput. Appl. Math..

[4]  Hua Zhou,et al.  Composition Markov chains of multinomial type , 2009, Advances in Applied Probability.

[5]  J. J. Foncannon.  Classical and quantum orthogonal polynomials in one variable , 2008 .

[6]  P. Diaconis,et al.  Gibbs sampling, exponential families and orthogonal polynomials , 2008, 0808.3852.

[7]  D. Bakry,et al.  The Hypergroup Property and Representation of Markov Kernels , 2006, math/0601605.

[8]  L. Brown,et al.  Interval Estimation for a Binomial Proportion , 2001 .

[9]  Dennis M. Healy,et al.  Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs , 1997, SIAM J. Comput..

[10]  S. Kocherlakota,et al.  Bivariate discrete distributions , 1992 .

[11]  Mizan Rahman,et al.  Cumulative Bernoulli trials and Krawtchouk processes , 1984 .

[12]  Robert C. Griffiths,et al.  ASPECTS OF CORRELATION IN BIVARIATE POISSON DISTRIBUTIONS AND PROCESSES , 1979 .

[13]  P. Diaconis.  Finite forms of de Finetti's theorem on exchangeability , 1977, Synthese.

[14]  J. B. Thomas,et al.  Characterization of a class of bivariate distribution functions , 1975 .

[15]  George Gasper,et al.  Banach Algebras for Jacobi Series and Positivity of a Kernel , 1972 .

[16]  D. Vere-Jones.  FINITE BIVARIATE DISTRIBUTIONS AND SEMIGROUPS OF NON-NEGATIVE MATRICES , 1971 .

[17]  B. C. Griffiths,et al.  Orthogonal Polynomials on the Multinomial Distribution , 1971 .

[18]  Robert C. Griffiths,et al.  The Canonical Correlation Coefficients of Bivariate Gamma Distributions , 1969 .

[19]  G. K. Eagleson,et al.  A CHARACTERIZATION THEOREM FOR POSITIVE DEFINITE SEQUENCES ON THE KRAWTCHOUK POLYNOMIALS1 , 1969 .

[20]  G. Eagleson.  Polynomial Expansions of Bivariate Distributions , 1964 .

[21]  S Bochner,et al.  Positive Zonal Functions on Spheres. , 1954, Proceedings of the National Academy of Sciences of the United States of America.

[22]  M. Manser,et al.  Chi-Squared Distribution , 2010 .

[23]  Persi Diaconis,et al.  Examples comparing importance sampling and the Metropolis algorithm , 2006 .

[24]  Patrick Suppes,et al.  When are probabilistic explanations possible? , 2005, Synthese.

[25]  W. Schoutens.  Stochastic processes and orthogonal polynomials , 2000 .

[26]  Patrick Suppes,et al.  A new proof of the impossibility of hidden variables using the principles of exchangeability and identity of conditional distributions , 1996 .

[27]  M. Kerimov.  The theory of error-correcting codes☆ , 1980 .

[28]  O. V. Sarmanov,et al.  Probabilistic Properties of Bilinear Expansions of Hermite Polynomials , 1967 .

[29]  M. Fréchet.  Sur les tableaux de correlation dont les marges sont donnees , 1951 .

[30]  A. C. Aitken,et al.  XI.—On Fourfold Sampling with and without Replacement , 1936 .

[31]  J. Meixner,et al.  Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion , 1934 .

[32]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .