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 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 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. Orthogonal
polynomial kernels are important to overcome some non-uniqueness difficulties
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 $d\toınfty.$
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