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  • 标题:Absolute continuity of the martingale limit in branching processes in random environment
  • 本地全文:下载
  • 作者:Ewa Damek ; Nina Gantert ; Konrad Kolesko
  • 期刊名称:Electronic Communications in Probability
  • 印刷版ISSN:1083-589X
  • 出版年度:2019
  • 卷号:24
  • DOI:10.1214/19-ECP229
  • 语种:English
  • 出版社:Electronic Communications in Probability
  • 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 $W$.
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