Open Access
June, 1969 On Branching Processes in Random Environments
Walter L. Smith, William E. Wilkinson
Ann. Math. Statist. 40(3): 814-827 (June, 1969). DOI: 10.1214/aoms/1177697589


$\{\zeta_n\}$ is a sequence of $\operatorname{iid}$ "environmental" variables in an abstract space $\Theta$. Each point $\zeta \varepsilon \Theta$ is associated with a $\operatorname{pgf} \phi_\zeta(s)$. The branching process $\{Z_n\}$ is defined as a Markov chain such that $Z_0 = k$, a finite integer, and given $Z_n$ and $\zeta_n, Z_{n+1}$ is distributed as the sum of $Z_n \operatorname{iid}$ random variables, each with $\operatorname{pgf} \phi_{\zeta_n}(s)$. Set $\xi(\zeta_n) = \phi'_{\zeta_n}(1)$ and assume that $E|\log\xi(\zeta_n)| < \infty$. Then: (i) $P\{Z_n = 0\} \rightarrow 1$ if $E \log \xi(\zeta_n) \leqq 0$; (ii) $qk = _{\operatorname{def}} \lim P\{Z_n = 0\} < 1$ if $E \log \xi(\zeta_n) > 0$ and $E|\log (1 - \phi_{\zeta_n}(0))| < \infty$. Furthermore $\{q_k\}, k = 1, 2, \cdots$, forms a moment sequence.


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Walter L. Smith. William E. Wilkinson. "On Branching Processes in Random Environments." Ann. Math. Statist. 40 (3) 814 - 827, June, 1969.


Published: June, 1969
First available in Project Euclid: 27 April 2007

zbMATH: 0184.21103
MathSciNet: MR246380
Digital Object Identifier: 10.1214/aoms/1177697589

Rights: Copyright © 1969 Institute of Mathematical Statistics

Vol.40 • No. 3 • June, 1969
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