On asymptotics of the probability of nonextinction for a critical branching process in random environment

For critical branching processes in random environment, under some slight restrictions, the following inequalities for the tails of the extinction time distribution are proved:
\begin{equation} c_1/\sqrt n\le\mathbf P(T>n)\le c_2/\sqrt n. \end{equation}

In the case of fractional linear generating functions, (1) is replaced by
$$ \mathbf P(T>n)\sim c/\sqrt n,\qquad n\to\infty. $$