Asymptotic behaviour of heavy-tailed branching processes in random environments

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 } $.