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

Let $(Z_n{)}_{n\ge 0}$ 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.