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Local Limit Approximations for Markov Population Processes

Part of: Distribution theory Markov processes

Published online by Cambridge University Press:  14 July 2016

Sanda N. Socoll  and
A.D. Barbour
Sanda N. Socoll*
Affiliation:
Universität Zürich
A.D. Barbour*
Affiliation:
Universität Zürich
*
∗Postal address: Angewandte Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
∗Postal address: Angewandte Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
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Abstract

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In this paper we are concerned with the equilibrium distribution ∏n of the nth element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a (2+α)th moment condition on the jump distributions, we establish a bound of order O(n-(α+1)/2√logn) on the difference between the point probabilities of ∏n and those of a translated Poisson distribution with the same variance. Except for the factor √logn, 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.

Keywords

Continuous-time Markov jump processequilibrium distributionpoint probabilitiesStein–Chen methodcoupling

MSC classification

Secondary: 60J75: Jump processes 62E17: Approximations to distributions (nonasymptotic)
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 690 - 708
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Work supported in part by Schweizerischer Nationalfonds Projekte Nrs 20-107935/1 and 20-117625/1.

References

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