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JOURNALS // Diskretnaya Matematika // Archive

Diskr. Mat., 2006 Volume 18, Issue 2, Pages 29–47 (Mi dm44)

This article is cited in 31 papers

On large deviations of branching processes in a random environment: a geometric distribution of the number of descendants

M. V. Kozlov


Abstract: A branching process $Z_n$ with geometric distribution of descendants in a random environment represented by a sequence of independent identically distributed random variables (the Smith–Wilkinson model) is considered. The asymptotics of large deviation probabilities $\boldsymbol{\mathsf P}(\ln Z_n>\theta n)$, $\theta>0$, are found provided that the steps of the accompanying random walk $S_n$ satisfy the Cramér condition. In the cases of supercritical, critical, moderate, and intermediate subcritical processes the asymptotics follow that of the large deviations probabilities $\boldsymbol{\mathsf P}(S_n\le\theta n)$. In strongly subcritical case the same asymptotics hold for $\theta$ greater than some $\theta^*$ (for $\theta\le\theta^*$ the asymptotics of large deviation probabilities are different).
This research was supported by the Russian Foundation for Basic Research, grant 04–01–00700, and by DFG, project 436 RUS 113/722.

UDC: 519.2

Received: 02.11.2004
Revised: 07.04.2006

DOI: 10.4213/dm44


 English version:
Discrete Mathematics and Applications, 2006, 16:2, 155–174

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