Branching processes in random environment die slowly

Let $Z_{n,}n=0,1,\mathrm{.}\mathrm{.}\mathrm{.},$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $% f_{0}(s),f_{1}(s),…,$andlet$S_{0}=0,S_{k}=X_{1}+…+X_{k},k\geq 1,$ be the associated random walk with $X_i=\log f_{i-1}^{\mathrm{\prime }}(1),$ $\tau (m,n)$ be the left-most point of minimum of $\{S_k,k\ge 0\}$ on the interval $[m,n],$ and $T=\min \{k:Z_k=0\}$. 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 $n\to \mathrm{\infty }$, 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 $nt\mathrm{.}$