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

Ewa Damek,Konrad Kolesko,Nina Gantert

doi:10.1214/19-ecp229

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 [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 $W$.

Highlights

We consider a supercritical branching process Zn in a stationary and ergodic random environment ξ =n≥0
Due to the martingale convergence theorem, it is known that the normalized population size Wn = Zn/(E[Zn|ξ]) converges almost surely to a random variable W
We prove that if W is not concentrated at 0 or 1 for almost every environment ξ the law of W conditioned on the environment ξ is absolutely continuous with a possible atom at 0

Summary

Introduction and statement of the main result

Due to the recursive equation (1.2) satisfied by W , see below, it is closely related to the local regularity for stationary solutions to affine type equations, see (1.3) below This motivated us to study absolute continuity of the law of W. Absolute continuity of the solution is much harder to prove if (A, C) does not possess a priori any regularity, as for instance in the case of Bernoulli convolutions A is concentrated at λ, for some 0 < λ < 1 and C is a Bernoulli random variable, i.e. C takes the values +1, −1 each with probability 1/2. In order to prove Theorem 1.2, we will need some additional statements provided

Further results

Integrability of ψ