## Electronic Journal of Probability

### Asymptotic behaviour of heavy-tailed branching processes in random environments

#### 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 $\bar{\xi }$.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 56, 17 pp.

Dates
Accepted: 29 April 2019
First available in Project Euclid: 5 June 2019

https://projecteuclid.org/euclid.ejp/1559700306

Digital Object Identifier
doi:10.1214/19-EJP311

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations

#### Citation

Hong, Wenming; Zhang, Xiaoyue. Asymptotic behaviour of heavy-tailed branching processes in random environments. Electron. J. Probab. 24 (2019), paper no. 56, 17 pp. doi:10.1214/19-EJP311. https://projecteuclid.org/euclid.ejp/1559700306

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