Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T13:57:18.926Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Barbour, A. D. (2008). Coupling a branching process to an infinite dimensional epidemic process. To appear in ESAIM Prob. Statist.Google Scholar
Barbour, A. D. and Jensen, J. L. (1989). Local and tail approximations near the Poisson limit. Scand. J. Statist. 16, 75–87.Google Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford University Press.Google Scholar
Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.Google Scholar
Chung, F. and Lu, L. (2006). Concentration inequalities and martingale inequalities: a survey. Internet Math. 3, 79–127.Google Scholar
Hamza, K. and Klebaner, F. C. (1995). Conditions for integrability of Markov chains. J. Appl. Prob. 32, 541–547.Google Scholar
Kurtz, T. G. (1981). Approximation of Population Processes (CBMS-NSF Regional Conf. Ser. Appl. Math. 36). SIAM, Philadelphia, PA.Google Scholar
Röllin, A. (2005). Approximation of sums of conditionally independent random variables by the translated Poisson distribution. Bernoulli 11, 1115–1128.CrossRefGoogle Scholar
Socoll, S. and Barbour, A. D. (2009). Translated Poisson approximation to equilibrium distributions of Markov population processes. To appear in Methodology Comput. Appl. Prob. Google Scholar