Rates of convergence of some multivariate Markov chains with polynomial eigenfunctions

TL;DR

This paper analyzes the convergence rates of various multivariate Markov chains with polynomial eigenfunctions using spectral methods, providing sharp nonasymptotic bounds for models in genetics, ecology, and statistical processes.

Contribution

It introduces a unified spectral approach to precisely quantify convergence rates for several multivariate Markov chains with polynomial eigenfunctions.

Findings

Derived sharp nonasymptotic convergence bounds

Applied spectral techniques to diverse models

Enhanced understanding of Markov chain mixing times

Abstract

We provide a sharp nonasymptotic analysis of the rates of convergence for some standard multivariate Markov chains using spectral techniques. All chains under consideration have multivariate orthogonal polynomial as eigenfunctions. Our examples include the Moran model in population genetics and its variants in community ecology, the Dirichlet-multinomial Gibbs sampler, a class of generalized Bernoulli--Laplace processes, a generalized Ehrenfest urn model and the multivariate normal autoregressive process.