Critical branching processes evolving in an unfavorable random environment

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 $\varphi (n)=o(a_{n})$.