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

TL;DR

This paper analyzes the asymptotic survival probability of a class of subcritical branching processes in random environments with heavy-tailed offspring distributions, establishing exponential decay with polynomial corrections and a Yaglom limit.

Contribution

It derives the asymptotic survival probability and a Yaglom type limit theorem for subcritical BPREs with heavy-tailed environmental effects, extending existing results to this class.

Findings

Survival probability decreases exponentially with polynomial correction.

Established a Yaglom type conditional limit theorem.

Analyzed the behavior of a conditioned heavy-tailed random walk.

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.