Absolute continuity of the martingale limit in branching processes in random environment
Absolute continuity of the martingale limit in branching processes in random environment
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DOI：
10.1214/19-ecp229
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发表时间：
2018-06
影响因子：
0.5
通讯作者：
E. Damek;N. Gantert;Konrad Kolesko
中科院分区：
文献类型：
--
作者：
E. Damek;N. Gantert;Konrad Kolesko
考虑了平稳遍历随机环境$\xi =（\xi_n）_{n\ge0}$中的超临界分支过程$Z_n$.由鞅收敛定理可知，归一化种群规模W_n=Z_n/（\mathbb E（Z_n|\xi））$几乎必然收敛于一个随机变量$W$。我们证明，如果$W$不集中在$0$或$1$，那么对于几乎每一个环境$\xi$，以环境$\xi $为条件的$W$的定律是绝对连续的，其中一个可能的原子在$0$。结果推广了\cite{Kaplan：1974}的主要结果，当然它也涵盖了Galton-Watson过程的鞅极限的著名情形.我们的证明结合了分析参数与递归描述的$W$。
We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $\xi =(\xi_n)_{n\ge0}$. 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 $W$.