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Diffusion approximations to linear stochastic difference equations with stationary coefficients

Published online by Cambridge University Press:  14 July 2016

Harry A. Guess  and
John H. Gillespie
Harry A. Guess
Affiliation:
National Institute of Environmental Health Sciences, National Institutes of Health
John H. Gillespie
Affiliation:
University of Pennsylvania
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Abstract

It is shown that solutions to linear first-order stochastic difference equations with stationary autocorrelated coefficients converge weakly in D[0,1] to an Ito stochastic integral plus a correction term when the time scale is shifted so that the means, variances, and covariances of the coefficients all approach zero at the same rate. Other limit theorems applicable to different time scale shifts are also given. These results yield two different continuous time limits to a recent model of Roughgarden (1975) for population growth in stationary random environments. One limit, an Ornstein-Uhlenbeck process, is applicable in the presence of rapidly fluctuating autocorrelated environments; the other limit, which is not a diffusion process, applies to the case of slowly varying, highly autocorrelated environments. Other applications in population biology and genetics are discussed.

Keywords

ITO STOCHASTIC INTEGRALSPOPULATION BIOLOGYPOPULATION GENETICSSTATIONARY PROCESSESSTOCHASTIC DIFFERENCE EQUATIONSLIMIT THEOREMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 1 , March 1977 , pp. 58 - 74
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Gillespie, J. H. and Guess, H. A. (1978) The effect of environmental autocorrelation on the progress of natural selection in random environments. Amer. Nat. To appear.Google Scholar
[3] Ibragimov, A. and Linnik, Y. V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen, The Netherlands.Google Scholar
[4] Khintchine, A. (1948) Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. Chelsea, New York.Google Scholar
[5] Kurtz, T. G. (1975) Semigroups of conditioned shifts and approximation of Markov processes. Ann. Prob. 3, 618–642.CrossRefGoogle Scholar
[6] Longnecker, M. and Serfling, R. J. (1976) A note on application of a lemma of Billingsley. Technical Report, Florida State University, Tallahassee, Florida.Google Scholar
[7] Natanson, I. P. (1955) Theory of Functions of a Real Variable, Volume 1. Ungar, New York.Google Scholar
[8] Norman, M. F. (1975) Diffusion approximation of non-Markovian processes. Ann. Prob. 3, 358–364.Google Scholar
[9] Roughgarden, J. (1975) A simple model for population dynamics in stochastic environments. Amer. Nat. 109, 713–736.Google Scholar
[10] Šidák, Z. (1967) Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 58, 626–633.Google Scholar
[11] Serfling, R. J. (1970) Moment inequalities for the maximum cumulative sum. Ann. Math. Statist. 41, 1227–1234.Google Scholar
[12] Stratonovich, R. L. (1966) A new representation for stochastic integrals. SIAM J. Control. 4, 362–371.Google Scholar
[13] Wong, E. and Zakaï, M. (1969) Riemann-Stieltjes approximations of stochastic integrals. Z. Wahrscheinlichkeitsth. 12, 87–97.Google Scholar