Absolute continuity of the martingale limit in branching processes in random environment

TL;DR

This paper proves that in supercritical branching processes within a stationary ergodic environment, the distribution of the normalized population size is absolutely continuous conditioned on the environment, except possibly at zero.

Contribution

It generalizes previous results by establishing absolute continuity of the martingale limit in a broad class of branching processes in random environments.

Findings

The law of the martingale limit W is absolutely continuous conditioned on the environment.

The result applies to supercritical processes in stationary ergodic environments.

It extends classical results for Galton-Watson processes.

Abstract

We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $\xi =(\xi_n)_{n\ge0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_n=Z_n/ (\mathbb E (Z_n|\xi ))$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $\xi$ the law of $W$ conditioned on the environment $\xi $ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of \cite{kaplan:1974}, and of course it covers the well-known case of the martingale limit of a Galton-Watson process. Our proof combines analytical arguments with the recursive description of $W$.