Limit theorems for weakly subcritical branching processes in random environment

TL;DR

This paper investigates the asymptotic behavior of weakly subcritical branching processes in random environments, revealing conditions under which the process exhibits supercritical characteristics when conditioned on survival.

Contribution

It introduces new functional limit theorems for conditional random walks and analyzes the survival probability and population size in weakly subcritical regimes.

Findings

Asymptotic survival probability characterized

Population size conditioned on non-extinction analyzed

Functional limit theorem established for conditional processes

Abstract

For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, 'supercritical'. This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on non-extinction. Also a functional limit theorem is proven, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.