On the Berry-Esseen Theorem for Random $U$-Statistics

Abstract

A Berry-Esseen theorem for $U$-statistics when the sample size is random is presented for the case when the random size is independent of the observations. This result extends the work of Callaert and Janssen. As an application of the special case of sample means, a rate of convergence to normality is obtained for the supercritical Galton-Watson process. Other possible applications are in sequential analysis.