Translated Poisson approximation to equilibrium distributions of Markov population processes

TL;DR

This paper establishes a translated Poisson distribution as a highly accurate approximation to the equilibrium distributions of density dependent Markov processes, improving understanding of their probabilistic behavior.

Contribution

It introduces a novel approximation method using translated Poisson distributions with strong total variation accuracy, leveraging Stein-Chen and Dynkin's formulas.

Findings

Approximation error in total variation norm is minimal and asymptotically optimal.

The translated Poisson distribution matches the mean and variance of the equilibrium distribution.

The method applies to continuous-time density dependent Markov processes.

Abstract

The paper is concerned with the equilibrium distributions of continuous-time density dependent Markov processes on the integers. These distributions are known typically to be approximately normal, and the approximation error, as measured in Kolmogorov distance, is of the smallest order that is compatible with their having integer support. Here, an approximation in the much stronger total variation norm is established, without any loss in the asymptotic order of accuracy; the approximating distribution is a translated Poisson distribution having the same variance and (almost) the same mean. Our arguments are based on the Stein-Chen method and Dynkin's formula.