Branching processes in random environment die slowly

TL;DR

This paper studies the long-term behavior of branching processes in random environments, revealing how their extinction times and population distributions depend on the associated random walk's properties.

Contribution

It provides new conditional limit theorems for the process's distribution, considering the position of the minimum of the associated random walk under the Doney condition.

Findings

Limit distributions depend on the location of the minimum of the associated random walk.

Conditional limit theorems are established for the process given extinction at time n.

The form of the limit distributions varies with the position of the minimum relative to nt.

Abstract

Let $Z_{n,}n=0,1,...,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $% f_{0}(s),f_{1}(s),...,$ and let $S_{0}=0,S_{k}=X_{1}+...+X_{k},k\geq 1,$ be the associated random walk with $X_{i}=\log f_{i-1}^{\prime}(1),$ $\tau (m,n)$ be the left-most point of minimum of $\left\{S_{k},k\geq 0\right\} $ on the interval $[m,n],$ and $T=\min \left\{k:Z_{k}=0\right\} $. Assuming that the associated random walk satisfies the Doney condition $P(S_{n}>0) \to \rho \in (0,1),n\to \infty ,$ we prove (under the quenched approach) conditional limit theorems, as $n\to \infty $, for the distribution of $Z_{nt},$ $Z_{\tau (0,nt)},$ and $Z_{\tau (nt,n)},$ $t\in (0,1),$ given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt.$