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 $E|logxi(zeta_n)| < infty$. Then: (i) $P{Z_n = 0} ightarrow 1$ 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.