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 ξ = (ξ n ) n≥0 . Due to the martingale convergence theorem, it is known that the normalized population size W n = Z n /(E(Z n |ξ)) 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 ξ the law of W conditioned on the environment ξ is absolutely continuous with a possible atom at 0. The result generalizes considerably the main result of [10], 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 .

Limit theorems for supercritical branching processes in random environment

“…[13] for the so called spectral Nagaev-Guivarc'h method). The proof of Proposition 2.1, contained in section 3, is inspired by the methods introduced in [3,8,11].…”