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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.

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
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2009
Deposited On:31 Dec 2009 10:16
Last Modified:05 Apr 2016 13:31
Publisher:Applied Probability Trust
ISSN:0021-9002
Publisher DOI:10.1239/jap/1253279846
Related URLs:http://projecteuclid.org/euclid.jap/1253279846
http://arxiv.org/abs/0902.0886v1
Permanent URL: http://doi.org/10.5167/uzh-23677

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