Local limit approximations for Markov population processes

TL;DR

This paper derives precise local limit approximations for the equilibrium distributions of density-dependent Markov processes, showing they closely resemble translated Poisson distributions with explicit error bounds.

Contribution

It introduces a novel bound on the difference between the equilibrium distribution and a translated Poisson approximation, using Stein-Chen method and coupling techniques.

Findings

Bound of order O(n^{-(α+1)/2}√log n) on probability differences

Approximation accuracy comparable to sums of independent variables

Applicable under a (2+α)-th moment condition on jumps

Abstract

The paper is concerned with the equilibrium distribution $\Pi_n$ of the $n$-th element in a sequence of continuous-time density dependent Markov processes on the integers. Under a $(2+\a)$-th moment condition on the jump distributions, we establish a bound of order $O(n^{-(\a+1)/2}\sqrt{\log n})$ on the difference between the point probabilities of $\Pi_n$ and those of a translated Poisson distribution with the same variance. Except for the factor $\sqrt{\log n}$, the result is as good as could be obtained in the simpler setting of sums of independent integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.