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 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 $\bar{\xi}$.