RC Griffiths and D Spano
n-Kernel Orthogonal Polynomials on the Dirichlet, Dirichlet-Multinomial, Poisson-Dirichlet and Ewens sampling distributions, and positive-definite sequences
Abstract: We consider a multivariate version of the so-called Lancaster problem of characterizing canonical correlation coe±cients of symmetric bivariate distributions with identical marginals and orthogonal polynomial expansions. The marginal distributions examined in this paper are the Dirichlet and the Dirichlet-Multinomial distribution, respectively on the continuous and the N-discrete d-dimensional simplex. Their infinite-dimensional limit distributions, respectively the Poisson-Dirichlet distribution and the Ewens' sampling formula, are considered as well. We study in particular the possibility of mapping canonical correlations on the d-dimensional continuous simplex (i) to canonical correlation sequences on the d + 1-dimensional simplex and/or (ii) to canonical correlations on the discrete simplex, and viceversa. Driven by this motivation, the ¯rst half of the paper is devoted to providing a full characterization and probabilistic interpretation of jnj-orthogonal polynomial kernels (i.e. sums of products of orthogonal polynomials of the same degree jnj) with respect to the mentioned marginal distributions. Orthogonal polynomial kernels are important to overcome some non-uniqueness di±culties arising when dealing with multivariate orthogonal (or bi-orthogonal) polynomials.We establish several identities and some integral representations which are multivariate extensions of important results known for the case d = 2 since the 1970's. These results, along with a common interpretation of the mentioned kernels in terms of dependent Polya urns, are shown to be key features leading to several non-trivial solutions to Lancaster's problem, many of which can be extended naturally to the limit as d ! 1: AMS 2000 subject classifications: 33C50, 60E05, 60G07.
Keywords: Multivariate orthogonal polynomials, Orthogonal polynomial kernels, Jacobi, Hahn, Dirichlet distribution, Dirichlet-Multinomial, Poisson-Dirichlet, Ewens' sampling formula, Canonical correlations, Positive-de¯nite sequences, Lancaster's problem.