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2019 Asymptotic behaviour of heavy-tailed branching processes in random environments
Wenming Hong, Xiaoyue Zhang
Electron. J. Probab. 24: 1-17 (2019). DOI: 10.1214/19-EJP311

Abstract

Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\mathbb{E} \log m(\xi _{0})=\infty $. We show that (1) there exists no proper $c_{n}$ such that $\{Z_{n}/c_{n}\}$ has a proper, non-degenerate limit; (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\left (\bar{\xi } ,Z_{n}(\bar{\xi } )\right )$ converges almost surely to a random variable $Y(\bar{\xi } )$, where $Y\in (0,1)~\eta $-a.s.; (3) finally, we give the necessary and sufficient conditions for the almost sure convergence of $\left \{\frac{U(\bar {\xi },Z_{n}(\bar {\xi }))} {c_{n}(\bar{\xi } )}\right \}$, where $U(\bar{\xi } )$ is a slowly varying function that may depend on $\bar{\xi } $.

Citation

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Wenming Hong. Xiaoyue Zhang. "Asymptotic behaviour of heavy-tailed branching processes in random environments." Electron. J. Probab. 24 1 - 17, 2019. https://doi.org/10.1214/19-EJP311

Information

Received: 31 October 2018; Accepted: 29 April 2019; Published: 2019
First available in Project Euclid: 5 June 2019

zbMATH: 07068787
MathSciNet: MR3968718
Digital Object Identifier: 10.1214/19-EJP311

Subjects:
Primary: 60J80
Secondary: 60F10

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