Path to survival for the critical branching processes in a random environment

TL;DR

This paper establishes a limit theorem for critical branching processes in random environments, showing that the scaled logarithm of the process converges to a Levy process conditioned to stay nonnegative, under specific conditions.

Contribution

It introduces a conditional functional limit theorem for critical branching processes in random environments, linking their scaled trajectories to Levy processes conditioned to remain nonnegative.

Findings

The scaled log process converges to a Levy process conditioned to stay nonnegative.

The proof utilizes a limit theorem for a driftless random walk conditioned to stay nonnegative.

The result applies under the conditions Z_n > 0 and p << n.

Abstract

A critical branching process $\left\{ Z_{k},k=0,1,2,...\right\} $ in a random environment is considered. A conditional functional limit theorem for the properly scaled process $\left\{ \log Z_{pu},0\leq u<\infty \right\} $ is established under the assumptions $Z_{n}>0$ and $p\ll n$. It is shown that the limiting process is a Levy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.