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

摘要

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: ? → ? 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.