Integro-local limit theorems for supercritical branching process in a random environment

Abstract

Let Zn be a supercritical branching process in a random environment (BPRE). Under certain moment assumptions, we present the precise asymptotics for the “integro-local” probabilities P(logZn∈[x(n),x(n)+Δn)), where Δn→0 and x(n)→∞ as n→∞. In particular, this implies the large deviations tail asymptotics for P(logZn⩾x(n)) as n→∞. Like in previous research, we can see that, in the light-tail case, the main term in the large deviations asymptotics for the BPRE is provided by the associated random walk.