Weighted moments of the limit of a branching process in a random environment

TL;DR

This paper establishes necessary and sufficient conditions for the existence of weighted moments of the limit of a supercritical branching process in a random environment, extending previous results in the Galton-Watson case.

Contribution

It provides a comprehensive characterization of weighted moments for the limit of branching processes in random environments, generalizing earlier findings.

Findings

Conditions for existence of weighted moments of W

Extension of results to general random environments

Improved understanding over classical Galton-Watson models

Abstract

Let $(Z_n)$ be a supercritical branching process in a random environment $% \zeta$, and $W$ be the limit of the normalized population size $Z_n/\mathbb{E%}(Z_n|\zeta)$. We show necessary and sufficient conditions for the existence of weighted moments of $W$ of the form $\E W^{\alpha}\ell(W)$, where $\alpha\geq 1$, $\ell$ is a positive function slowly varying at $\infty$. In the Galton-Watson case, the results improve those of Bingham and Doney (1974).