Theorems on large deviations for randomly idexed sum of weighted random variables

Abstract

In this paper, we consider a random variable Zt = Nt i=1 aiXi, where X,X1,X2, . . . are independent identically distributed random variables with mean EX = μ and variance DX = σ2 > 0. It is assumed that Z0 = 0, 0 ≤ ai <∞, and Nt , t ≥ 0 is a non-negative integervalued random variable independent of Xi , i = 1, 2, . . . . The paper is devoted to the analysis of accuracy of the standard normal approximation to the sum ˜Zt = (DZt) −1/2(Zt− EZt), large deviation theorems in the Cramer and power Linnik zones, and exponential inequalities for P (˜Zt ≥ x).