Conditional limit theorems for intermediately subcritical branching processes in random environment

TL;DR

This paper investigates the asymptotic behavior of intermediately subcritical branching processes in random environments, revealing a phase transition, survival probability decay, and alternating population sizes conditioned on non-extinction.

Contribution

It characterizes the phase transition and bottleneck behavior specific to the intermediately subcritical regime in branching processes with random environments.

Findings

Survival probability exhibits a specific asymptotic decay.

Population sizes alternate between small and large during non-extinction periods.

Bottleneck behavior is unique to the intermediately subcritical case.

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. For the subcritical regime a kind of phase transition appears. In this paper we study the intermediately subcritical case, which constitutes the borderline within this phase transition. We study the asymptotic behavior of the survival probability. Next the size of the population and the shape of the random environment conditioned on non-extinction is examined. Finally we show that conditioned on non-extinction periods of small and large population sizes alternate. This kind of 'bottleneck' behavior appears under the annealed approach only in the intermediately subcritical case.