Limit theorems for critical branching processes in an extremely unfavorable random environment

Abstract

Let $\{Z_m,m\ge 0\}$ be a critical branching process in random environment and $\{S_m,m\ge 0\}$ be its associated random walk. Assuming that the increments distribution of the associated random walk belongs without centering to the domain of attraction of an $\alpha$-stable law we prove conditional limit theorems describing, as $n\to \infty$, the distribution the number of particles in the process $\{Z_m,0\le m\le n\}$ given $Z_n>0$ and $S_n\le const$.

26 pages