Branching processes in random environment die slowly

Abstract

Let $Z_{n,}n=0,1,...,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions and let $S_0=0,S_k=X_1+...+X_k,k\ge 1,$ be the associated random walk with $X_i=\log f_{i-1}^{\prime }(1),$ $\tau (m,n)$ be the left-most point of minimum of $\left\{S_k,k\ge 0\right\}$ on the interval $[m,n],$ and $T=\min \left\{k:Z_k=0\right\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n>0)\to \rho \in (0,1),n\to \infty ,$ we prove (under the quenched approach) conditional limit theorems, as $n\to \infty$, for the distribution of $Z_{nt},$ $Z_{\tau (0,nt)},$ and $Z_{\tau (nt,n)},$ $t\in (0,1),$ given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt.$