Self-normalized Cramer moderate deviations for a supercritical Galton-Watson process

TL;DR

This paper derives optimal or near-optimal self-normalized Cramér moderate deviation results and Berry-Esseen bounds for the Lotka-Nagaev estimator in supercritical Galton-Watson processes, enhancing understanding of its probabilistic behavior.

Contribution

It provides the first self-normalized Cramér moderate deviation and Berry-Esseen bounds for the Lotka-Nagaev estimator in supercritical Galton-Watson processes.

Findings

Established near-optimal moderate deviation bounds

Derived Berry-Esseen bounds for the estimator

Enhanced probabilistic understanding of the estimator's behavior

Abstract

Let $(Z_n)_{n\geq0}$ be a supercritical Galton-Watson process. Consider the Lotka-Nagaev estimator for the offspring mean. In this paper, we establish self-normalized Cram\'{e}r type moderate deviations and Berry-Esseen's bounds for the Lotka-Nagaev estimator. The results are believed to be optimal or near optimal.