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

Electronic Communications in Probability

Ewa Damek¹ 

关键词: branching processes;    branching processes in random environment;    martingale limit;   

DOI  :  10.1214/19-ECP229

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

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 $W$.

【 授权许可】

CC BY