On the rate of convergence in the global central limit theorem for random sums of independent random variables

Abstract

We present upper bounds of the integral \( {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathbf{P}\left\{{Z}_N 0 (S N = X 1 + · · · + X N ) of centered independent random variables X 1 ,X 2 , . . . . The number of summands N is a nonnegative integer-valued random variable independent of X 1 ,X 2 , . . . .