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 log A⁻¹ (1 − A) is regularly varying with a parameter α > 1, that is that P (log A⁻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 (Z_l ≥ m) of lth population Z_l is of order (log^(l) m)^−α L (log^(l) m) for large m, where log^(l) m = log…log m. The proof is mainly based on Tauberian theorem. Using this result we also analyze the asymptotic behavior of the first passage time T_n of the state n ∈ Z by the walker in a neighborhood random walk in random environment created by independent copies (A_i: i ∈ Z) of (0, 1)-valued random variable A.