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

Motivated by seminal paper of Kozlov 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 $\log A^{-1} (1-A)$ is regularly varying with a parameter $\alpha>1$, that is that ${\bf P} \Big( \log A^{-1} (1-A) > x \Big) = x^{-\alpha} L(x)$ 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 $n$-th generation which gets even heavier with $n$ increasing. Precisely, in this work, we prove that asymptotic tail ${\bf P}(Z_l \ge m)$ of $l$-th population $Z_l$ is of order $\Big(\log^{(l)} m \Big)^{-\alpha} L \Big(\log^{(l)} m \Big)$ for large $m$, where $\log^{(l)} m = \log \ldots \log m$. The proof is mainly based on Tauberian theorem. Using this result we also analyze the asymptotic behaviour of the first passage time $T_n$ of the state $n \in \mathbb{Z}$ by the walker in a neighborhood random walk in random environment created by independent copies $(A_i : i \in \mathbb{Z})$ of $(0,1)$-valued random variable $A$. This version differs from the final version as it contains an alternative proof for the tail behavior for generation sizes which is not very sharp (lacks constant) but completely avoids arguments based on Tauberian theorem. This proof may be of an independent interest.