Berry-esseen bounds for random sums of non-identically distributed random variables

Abstract

Let X1,X2, . . . be independent random variables with mean zero and finite third moment and N a positive integral-valued random variable such that N,X1, X2, . . . are independent. In this work, we use a concentration inequality approach of Stein’s method to establish uniform and non-uniform Berry-Esseen bounds for random sums X1 +X2 + · · · +XN in the case that X1,X2, . . . are not necessarily identically distributed random variables.