Some estimates of normal approximation for the distribution of a sum of a random number of independent random variables

Abstract

In this paper, we generalize one our result for a fixed number of summands of independent, not necessarily identically distributed, real random variables in [J. Sunklodas, Some estimates of the normal approximation for independent nonidentically distributed random variables, Lith. Math. J., 51(1):66-74, 2011] to the case of a random number of summands. We estimate the quantity {pipe}Eh(Z N) - Eh(Y) {pipe} when h: ℝ → ℝ is a real thrice-differentiable function, S N = X 1...+... X N, Z N = (S N ES N)/√DS N) where X 1,X 2, . . . are independent, not necessarily identically distributed, real random variables, N is a positive integer-valued r. v. independent of X 1, X 2, . . ., and Y is a standard normal random variable. © 2012 Springer Science+Business Media, Inc.