Electronic Communications in Probability

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

Ewa Damek, Nina Gantert, and Konrad Kolesko

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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 $W$.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 42, 13 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1562119370

Digital Object Identifier
doi:10.1214/19-ECP229

Zentralblatt MATH identifier
07088983

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments

Keywords
branching processes branching processes in random environment martingale limit

Rights
Creative Commons Attribution 4.0 International License.

Citation

Damek, Ewa; Gantert, Nina; Kolesko, Konrad. Absolute continuity of the martingale limit in branching processes in random environment. Electron. Commun. Probab. 24 (2019), paper no. 42, 13 pp. doi:10.1214/19-ECP229. https://projecteuclid.org/euclid.ecp/1562119370


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