On the survival probability for a class of subcritical branching processes in random environment

Abstract

Let $Z_{n}$ be the number of individuals in a subcritical Branching Process in Random Environment (BPRE) evolving in the environment generated by i.i.d. probability distributions. Let $X$ be the logarithm of the expected offspring size per individual given the environment. Assuming that the density of $X$ has the form \[p_{X}(x)=x^{-\beta-1}l_{0}(x)e^{-\rho x}\] for some $\beta>2$, a slowly varying function $l_{0}(x)$ and $\rho\in(0,1)$, we find the asymptotic of the survival probability $\mathbb{P}(Z_{n}>0)$ as $n\rightarrow\infty$, prove a Yaglom type conditional limit theorem for the process and describe the conditioned environment. The survival probability decreases exponentially with an additional polynomial term related to the tail of $X$. The proof uses in particular a fine study of a random walk (with negative drift and heavy tails) conditioned to stay positive until time $n$ and to have a small positive value at time $n$, with $n\rightarrow\infty$.