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Asymptotic behaviour of heavy-tailed branching processes in random environments. (English) Zbl 1466.60178
Summary: 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., \(y_{n} (\bar{\xi} ,Z_{n}(\bar{\xi}))\) 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 \(\{\frac{U(\bar {\xi},Z_{n}(\bar {\xi}))} {c_{n}(\bar{\xi})}\}\), where \(U(\bar{\xi})\) is a slowly varying function that may depend on \(\bar{\xi}\).
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F10 Large deviations
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