Galton–Watson branching processes in a random environment. I: limit theorems

Let $Z_n$ be the number of individuals at time $n$ in a branching process in a random environment generated by independent identically distributed random probability generating functions $f_0(s),f_1(s),\dots,f_n(s),\dots$ . Let
$$ X_i=\log f_{i-1}'(1),\qquad i=0,1,2,\dots; \qquad S_0=0,\quad S_n=X_1+\dots+X_n,\qquad n\ge 1. $$
It is shown that if $Z_n$ is, in a sense, “critical,” then there exists a limit in distribution
$$ \lim_{n\to\infty}\exp\Bigl\{-\min_{0\le j\le n}S_j\Bigr\}\,\mathbf{P}\{Z_n>0\mid f_0,\dots,f_{n-1}\}=\zeta, $$
where $\zeta$ is a proper random variable positive with probability 1. In addition, it is shown that for a “typical” realization of the environment the number of individuals $Z_n$ given $\{Z_n>0\}$ grows as $\exp\{S_n-\min_{0\le j\le n}S_j\}$ (up to a positive finite random multiplier).