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Equilibrium distributions Markov population processes

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour
A. D. Barbour*
Affiliation:
Gonville and Caius College, Cambridge
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Abstract

Distributional limit theorems, together with rates of convergence, are obtained for the equilibrium distributions of a wide variety of one-dimensional Markov population processes. Three separate cases are considered. First, in the standard setting, the convergence as N→∞ of √N(xN-c) to a normal distribution is established, together with a rate of convergence of O(N−1/2), under weaker conditions than those previously imposed: here, c represents the unique equilibrium of the deterministic equations ẋ = F(x), and xN denotes the population process under its equilibrium distribution. This convergence holds if F′(c)<0: the next section shows that, if F′(c) = 0, both the normalization and the limit distribution are different. Finally, sequences of processes xN suitable for approximating genetical models are considered. In these circumstances, xN itself converges in distribution as N→∞, and the convergence rate is essentially O(N-1), though modification is sometimes needed near natural boundaries.

Keywords

DENSITY-DEPENDENT MARKOV POPULATION PROCESSEQUILIBRIUM DISTRIBUTIONRATES OF CONVERGENCE
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 3 , September 1980 , pp. 591 - 614
Copyright
Copyright © Applied Probability Trust 1980 

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References

Alm, S. E. (1978) On the Rate of Convergence in Diffusion Approximation of Jump Markov Processes. Ph.D. Thesis, Uppsala University.Google Scholar
Barbour, A. D. (1975) The duration of the closed stochastic epidemic. Biometrika 62, 477–482.Google Scholar
Barbour, A. D. (1976) Quasi-stationary distributions in Markov population processes. Adv. Appl. Prob. 8, 296–314.Google Scholar
Dynkin, E. B. (1965) Markov Processes. Springer-Verlag, Berlin.Google Scholar
Ellis, R. S. and Newman, C. M. (1978) Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrscheinlichkeitsth. 44, 117–139.Google Scholar
Hartman, P. (1964) Ordinary Differential Equations. Wiley, New York.Google Scholar
Kurtz, T. G. (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Prob. 7, 49–58.CrossRefGoogle Scholar
Kurtz, T. G. (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Prob. 8, 344–356.Google Scholar
Kurtz, T. G. (1976) Limit theorems and diffusion approximations for density dependent Markov chains. Math. Programming Study 5, 67–78.Google Scholar
Kurtz, T. G. (1978) Strong approximation theorems for density dependent Markov chains. Stoch. Proc. Appl. 6, 223–240.Google Scholar
Norman, M. F. (1972) Markov Processes and Learning Models. Academic Press, New York.Google Scholar
Norman, M. F. (1974) A central limit theorem for Markov processes that move by small steps. Ann. Prob. 2, 1065–1074.Google Scholar
Osei, G. K. and Thompson, J. W. (1977) The supersession of one rumour by another. J. Appl. Prob. 14, 127–134.CrossRefGoogle Scholar
Ridler-Rowe, C. J. (1978) On competition between two species. J. Appl. Prob. 15, 457–465.Google Scholar
Stanley, H. E. (1971) Phase Transitions and Critical Phenomena. Clarendon Press, Oxford.Google Scholar
Stein, C. (1970) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. 6th Berk. Symp. Math. Statist. Prob. 2, 583–602.Google Scholar