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

Abstract

Let $Z_n$ be the number of individuals in a subcritical BPRE evolving in the environment generated by iid 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 $\rho \in \left(0,1\right),$ we find the asymptotic survival probability and prove a Yaglom type conditional limit theorem for the process. The survival probability decreases exponentially with an additional polynomial term related to the tail of $X$. The proof relies on 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 $n$ tending to infinity.

References in corpus (1)

• Subcritical branching processes in random environment without Cramer condition

Cited by in corpus (1)

• Random walk with heavy tail and negative drift conditioned by its minimum and final values