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2019 Absolute continuity of the martingale limit in branching processes in random environment
Ewa Damek, Nina Gantert, Konrad Kolesko
Electron. Commun. Probab. 24: 1-13 (2019). 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 $W$.

Citation

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Ewa Damek. Nina Gantert. Konrad Kolesko. "Absolute continuity of the martingale limit in branching processes in random environment." Electron. Commun. Probab. 24 1 - 13, 2019. https://doi.org/10.1214/19-ECP229

Information

Received: 21 June 2018; Accepted: 10 April 2019; Published: 2019
First available in Project Euclid: 3 July 2019

zbMATH: 07088983
MathSciNet: MR3978691
Digital Object Identifier: 10.1214/19-ECP229

Subjects:
Primary: 60J80, 60K37

JOURNAL ARTICLE
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