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A limit theorem for discrete-parameter random evolutions

Published online by Cambridge University Press:  01 July 2016

Gilles A. Blum
Gilles A. Blum*
Affiliation:
Case Western Reserve University
*
∗Postal address: Department of Mathematics and Statistics, Case Western Reserve University, University Circle, Cleveland, OH 44106, U.S.A.
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Abstract

In this paper we obtain a limit theorem for discrete-parameter random evolutions. This theorem is then used to obtain diffusion approximations to the Wright-Fisher model in a Markovian environment and to sequences of stochastic difference equations.

Keywords

DIFFUSION APPROXIMATIONWEAK CONVERGENCESTOCHASTIC DIFFERENCE EQUATION
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 2 , June 1984 , pp. 281 - 292
Copyright
Copyright © Applied Probability Trust 1984 

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References

Blumenthal, R. M. and Getoor, R. K. (1968) Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Gillespie, J. and Guess, H. (1978) The effects of environmental autocorrelations on the progress of selection in a random environment. Amer. Naturalist 112, 897–909.CrossRefGoogle Scholar
Karun, S. and Levikson, B. (1974) Temporal fluctuations in selection intensities; case of small population size. Theoret. Popn Biol. 6, 383–412.Google Scholar
Kurtz, T. G. (1969) Extensions of Trotter's operator semigroup approximation theorems. J. Functional Anal. 3, 111–132.CrossRefGoogle Scholar
Kurtz, T. G. (1973) A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Functional Anal. 12, 55–67.CrossRefGoogle Scholar
Kurtz, T. G. (1975) Semigroups of conditioned shifts and approximation of Markov processes. Ann. Prob. 4, 618–642.Google Scholar
Kushner, H. J. (1980) A martingale method for the convergence of a sequence of processes to a jump diffusion processes. Z. Wahrscheinlichkeitsth. 53, 207–219.CrossRefGoogle Scholar
O'Brien, G. L. (1974) Limit theorems for sums of chain dependent processes. J. Appl. Prob. 11, 582–587.Google Scholar
Papanicolaou, G. C., Stroock, D. and Varadhan, S. R. S. (1977) Martingale approach to some limit theorems. In Statistical Mechanics and Dynamical Systems, by D. Ruelle and Papers from the 1976 Turbulence Conference, Duke University, Durham, NC.Google Scholar