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August 2022 Limit theorems for supercritical branching processes in random environment
Dariusz Buraczewski, Ewa Damek
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Bernoulli 28(3): 1602-1624 (August 2022). DOI: 10.3150/21-BEJ1349

Abstract

The branching process in random environment {Zn}n0 is a population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with positive probability and grows exponentially fast on the nonextinction set. Using Fourier techniques we improve existing central limit theorem as well as we obtain Edgeworth expansions and the renewal theorem for the sequence {logZn}n0. The strategy is to compare logZn with partial sums of i.i.d. random variables in order to obtain precise estimates.

Acknowledgement

We thank two anonymous referees for careful reading and numerous helpful suggestions. We are also grateful to Piotr Dyszewski for useful comments and discussions. Ewa Damek incorporated some ideas communicated her by Konrad Kolesko during the work on [8]. The research was partially supported by the National Science Center, Poland (grant number 2019/33/ B/ST1/00207).

Citation

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Dariusz Buraczewski. Ewa Damek. "Limit theorems for supercritical branching processes in random environment." Bernoulli 28 (3) 1602 - 1624, August 2022. https://doi.org/10.3150/21-BEJ1349

Information

Received: 1 July 2020; Published: August 2022
First available in Project Euclid: 25 April 2022

Digital Object Identifier: 10.3150/21-BEJ1349

Keywords: Berry Esseen bound , branching process , central limit theorem , Characteristic function , Edgeworth expansions , Fourier transform , random environment , Renewal theorem

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Vol.28 • No. 3 • August 2022
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