On Branching Processes in Random Environments

Abstract

$\{\zeta_n\}$ 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.