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→∞.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ403 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
