Precise large deviation estimates for branching process in random environment

TL;DR

This paper derives precise asymptotic estimates for large deviations in a branching process within a random environment, including upper deviations and first passage times under certain conditions.

Contribution

It provides new asymptotic formulas for large deviations and passage times in branching processes with random environments, extending understanding of their tail behaviors.

Findings

Derived precise asymptotics for upper large deviations.

Established large deviation estimates for first passage times.

Analyzed the total population size in subcritical cases.

Abstract

We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, 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 Cram\'er 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.