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

Abstract

We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $\xi =({\xi }_n{)}_{n\ge 0}$. 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 $W$.