Branching processes in random environment which extinct at a given moment

Abstract

Let ${Z_{n},n\geq 0} $ be a critical branching process in random environment and let $T$ be its moment of extinction. Under the annealed approach we prove, as $n\to \infty ,$ a limit theorem for the number of particles in the process at moment $n$ given $T=n+1$ and a functional limit theorem for the properly scaled process ${Z_{nt},\delta \leq t\leq 1-\delta} $ given $T=n+1$ and $\delta \in (0,1/2)$.