# Local limit approximations for Markov population processes

Barbour, A D; Socoll, S (2009). Local limit approximations for Markov population processes. Journal of Applied Probability, 46(3):690-708.

## Abstract

We are concerned with the equilibrium distribution $\prod _n$ of the $n$th element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a $(2+\alpha )$th moment condition on the jump distributions, we establish a bound of order $O(n^{-(\alpha +1)/2}\sqrt{ \log n})$ on the difference between the point probabilities of $\prod n$ 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.

## Abstract

We are concerned with the equilibrium distribution $\prod _n$ of the $n$th element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a $(2+\alpha )$th moment condition on the jump distributions, we establish a bound of order $O(n^{-(\alpha +1)/2}\sqrt{ \log n})$ on the difference between the point probabilities of $\prod n$ 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.

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## Additional indexing

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2009 31 Dec 2009 10:16 06 Dec 2017 21:37 Applied Probability Trust 0021-9002 https://doi.org/10.1239/jap/1253279846 http://projecteuclid.org/euclid.jap/1253279846http://arxiv.org/abs/0902.0886v1

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