| Autor: |
Damek, Ewa, Gantert, Nina, Kolesko, Konrad |
| Rok vydání: |
2018 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
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 $W$. |
| Databáze: |
arXiv |
| Externí odkaz: |
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