Slower variation of the generation sizes induced by heavy-tailed environment for geometric branching

Motivated by seminal paper of Kozlov et al. Kesten et al. (1975) we consider in this paper a branching process with a geometric offspring distribution parametrized by random success probability A and immigration equals 1 in each generation. In contrast to above mentioned article, we assume that environment is heavy-tailed, that is logA−1(1−A) is regularly varying with a parameter α>1, that is that P(logA−1(1−A)>x)=x−αL(x) for a slowly varying function L. We will prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of distribution of the population at nth generation which gets even heavier with n increasing. Precisely, in this work, we prove that asymptotic tail P(Zl≥m) of lth population Zl is of order (log(l)m)−αL(log(l)m) for large m, where log(l)m=log…logm. The proof is mainly based on Tauberian theorem. Using this result we also analyze the asymptotic behavior of the first passage time Tn of the state n∈Z by the walker in a neighborhood random walk in random environment created by independent copies (Ai:i∈Z) of (0,1)-valued random variable A.