Critical branching processes in random environment: the probability of extinction at a given moment

Let $\kappa$ be the extinction moment of a critical branching process in random environment with linear-fractional generating functions $f_n(s)$ of the number of particles in the $n$th generation, $n=1,2,\dots$ We give conditions on the parameters of $f_n(s)$ under which
$$ \mathsf P\{\kappa=n\}\sim cn^{-3/2}, $$
as $n\to\infty$, where $c$ is a positive constant.
This research was supported by the Russian Foundation for Basic Research, grant 96–01–00338, and by INTAS–RFBR, grant 95–0099.