Lancaster distributions and Markov chains with Multivariate Poisson-Charlier, Meixner and Hermite-Chebycheff polynomial eigenfunctions

TL;DR

This paper introduces new characterizations of multivariate distributions using orthogonal polynomials and explores Markov chains with these polynomials as eigenfunctions, extending classical results.

Contribution

It extends classical Lancaster characterizations to multivariate distributions with orthogonal polynomial expansions and constructs new Markov chains with these polynomials as eigenfunctions.

Findings

New Lancaster characterizations for multivariate distributions

Construction of Markov chains with polynomial eigenfunctions

Identification of mixture models with fixed stationary distributions

Abstract

This paper studies new Lancaster characterizations of bivariate multivariate Poisson, negative binomial and normal distributions which have diagonal expansions in multivariate orthogonal polynomials. The characterizations extend classical Lancaster characterizations of bivariate 1-dimensional distributions. Multivariate Poisson-Charlier, Meixner and Hermite-Chebycheff orthogonal polynomials, used in the characterizations, are constructed from classical 1-dimensional orthogonal polynomials and multivariate Krawtchouk polynomials. New classes of transition functions of discrete and continuous time Markov chains with these polynomials as eigenfunctions are characterized. The characterizations obtained belong to a class of mixtures of multi-type birth and death processes with fixed multivariate Poisson or multivariate negative binomial stationary distributions.