Geometric rate of growth in markov chains with applications to population-size-dependent models with dependent offspring

Harry Cohn, Fima Klebaner

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This paper studies the limit behaviour of {(Zn − an)/bn} where {Zn} is a real-valued temporally homogeneous Markov chain, and {an} and {bn} are some constants; the results are then applied to a general population model. In such a model Zn represents the nth generation population size and is defined as [formulla omitted] where {Xi n-1} are the offspring variables of the (n−1)th generation which are assumed to depend on n, i and Zn−1’ whereas the classical conditional independence of {xi n, i=1,…,Zn} given Zn is superseded by milder assumptions. Some necessary and sufficient conditions for {Zn/bn} to converge a.s. are derived, and some results on the robustness of the asymptotic behaviour of the Galton-Watson process are obtained when offspring independence is relaxed.

Original languageEnglish
Pages (from-to)283-307
Number of pages25
JournalStochastic Analysis and Applications
Issue number3
Publication statusPublished - 1 Jan 1986
Externally publishedYes

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