Multivariate approximation in total variation, I: equilibrium distributions of Markov jump processes

TL;DR

This paper develops a multivariate approximation theory using Stein's method, focusing on equilibrium distributions of Markov jump processes, extending univariate translated Poisson approximations to higher dimensions.

Contribution

It introduces a new multivariate approximation framework based on equilibrium distributions of Markov jump processes, generalizing univariate Poisson approximations.

Findings

Provides total variation error bounds for Markov jump process approximations

Extends Stein's method to multivariate equilibrium distributions

Lays groundwork for discrete normal approximation in higher dimensions

Abstract

For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein--Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein's method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimension are the immigration--death processes with Poisson distributions as equilibria. The method is illustrated by providing total variation error bounds for the approximation of the equilibrium distribution of one Markov jump process by that of another. In a companion paper, it is shown how to use the method for discrete normal approximation in ${\mathbb Z}^d$.