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May 2013 Orthogonal polynomial kernels and canonical correlations for Dirichlet measures
Robert C. Griffiths, Dario Spanò
Bernoulli 19(2): 548-598 (May 2013). DOI: 10.3150/11-BEJ403

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 Pólya 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 $d\rightarrow\infty$.

Citation

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Robert C. Griffiths. Dario Spanò. "Orthogonal polynomial kernels and canonical correlations for Dirichlet measures." Bernoulli 19 (2) 548 - 598, May 2013. https://doi.org/10.3150/11-BEJ403

Information

Published: May 2013
First available in Project Euclid: 13 March 2013

zbMATH: 1281.60015
MathSciNet: MR3037164
Digital Object Identifier: 10.3150/11-BEJ403

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability

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Vol.19 • No. 2 • May 2013
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