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 a 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 depends on $\bar{\xi }$.