  Let $\left\{ Z_{n},n=0,1,2,...\right\} $ be a critical branching process in
random environment and let $\left\{ S_{n},n=0,1,2,...\right\} $ be its
associated random walk. It is known that if the increments of this random walk
belong (without centering) to the domain of attraction of a stable law, then
there exists a sequence $a_{1},a_{2},...,$ slowly varying at infinity such that
the conditional distributions \begin{equation*} \mathbf{P}\left(
\frac{S_{n}}{a_{n}}\leq x\Big|Z_{n}>0\right) ,\quad x\in (-\infty ,+\infty ),
\end{equation*}% weakly converges, as $n\rightarrow \infty $ to the
distribution of a strictly positive and proper random variable. In this paper
we supplement this result with a description of the asymptotic behavior of the
probability \begin{equation*} \mathbf{P}\left( S_{n}\leq \varphi
(n);Z_{n}>0\right) , \end{equation*}% if $\varphi (n)\rightarrow \infty $ \ as
$n\rightarrow \infty $ in such a way that $\varphi (n)=o(a_{n})$.
