Asymptotic of the distribution and harmonic moments for a supercritical branching process in a random environment

Abstract

Let $(Z_n)$ be a supercritical branching process in an independent and identically distributed random environment $\xi$. We show the exact decay rate of the probability $\mathbb{P}(Z_n=j|Z_0=k)$ as $n\to \infty$, for each $j\ge k,$ assuming that $\mathbb{P}(Z_1=0)=0$. We also determine the critical value for the existence of harmonic moments of the random variable $W={\lim }_{n\to \infty }\frac{Z_n}{\mathbb{E}(Z_n|\xi )}$ under a simple moment condition.

References in corpus (1)

• Berry-Esseen's bound and Cramér's large deviation expansion for a supercritical branching process in a random environment

Cited by in corpus (1)

• Random walks and branching processes in correlated Gaussian environment