Critical branching processes evolving in an unfavorable random environment

Let $\{Z_n,n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ be a critical branching process in random environment and let $\{S_n,n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ 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 $a_1,a_2,\mathrm{.}\mathrm{.}\mathrm{.},$ 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 $n\to \mathrm{\infty }$ 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 $\phi (n)\to \mathrm{\infty }$ \ as $n\to \mathrm{\infty }$ in such a way that $\phi (n)=o(a_n)$.