Precise large deviation estimates for branching process in random environment

Abstract

We consider the branching process in random environment $\{Z_n{\}}_{n\ge 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We describe precise asymptotics of upper large deviations, i.e. $\mathbb{P}[Z_n>e^{\rho n}]$. Moreover in the subcritical case, under the Cramér condition on the mean of the reproduction law, we investigate large deviations-type estimates for the first passage time of the branching process in question and its total population size.

28 pages