Orthogonal polynomial kernels and canonical correlations for Dirichlet measures

TL;DR

This paper characterizes orthogonal polynomial kernels and explores their role in understanding canonical correlations for Dirichlet measures and their limits, providing new identities and interpretations in multivariate settings.

Contribution

It offers a complete characterization of n-orthogonal polynomial kernels for Dirichlet distributions and extends classical results to multivariate and infinite-dimensional cases.

Findings

Derived identities and integral representations for polynomial kernels

Established probabilistic interpretations via dependent Polya urns

Extended solutions to Lancaster's problem to high-dimensional and limit distributions

Abstract

We consider a multivariate version of the so-called Lancaster problem of characterizing canonical correlation coefficients 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 Ewens's 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 vice versa. Driven by this motivation, the first half of the paper is devoted to providing a full characterization and probabilistic interpretation of n-orthogonal polynomial kernels (i.e., sums of products of orthogonal polynomials of the same degree n) with respect to the mentioned marginal distributions. We establish several identities and some integral representations which are multivariate extensions of important results known for the case d=2 since the 1970s. 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 $d\rightarrow\infty$.