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Limit theorems for stochastic growth models. II

Published online by Cambridge University Press:  01 July 2016

Harry Kesten
Harry Kesten*
Affiliation:
Cornell University
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Abstract

We consider d-dimensional stochastic processes which take values in (R+)d These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some

Here τ:(R+)d→R +, |x| = σ1d |x(i)|, A {x ∈(R+)d: |x| 1} and T: A→A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Zn|ρnwp. This result is a generalization of the main limit theorem for supercritical branching processes; note, however, that in the present situation both ρ and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.

Keywords

GROWTH MODELSPOPULATION GROWTHEXPONENTIAL GROWTHMALTHUSIAN PARAMETERGROWTH RATEFIXED POINTPOPULATION COMPOSITIONCONVERGENCE OF COMPOSITION OF POPULATIONFISHER-WRIGHT-HALDANE MODEL
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 3 , December 1972 , pp. 393 - 428
Copyright
Copyright © Applied Probability Trust 1972 

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