Theorems on Large Deviations for Randomly Indexed Sum of Weighted Random Variables

In this paper, we consider a random variable $Z_{t}=\sum_{i=1}^{N_{t}}a_{i}X_{i}$ , where $X, X_{1}, X_{2}, \ldots$ are independent identically distributed random variables with mean E X=μ and variance D X=? 2>0. It is assumed that Z 0=0, 0?a i negative integer-valued random variable independent of X i , i=1,2,??. The paper is devoted to the analysis of accuracy of the standard normal approximation to the sum $\tilde{Z}_{t}=(\mathbf{D}Z_{t})^{-1/2}(Z_{t}-\mathbf{E}Z_{t})$ , large deviation theorems in the Cramer and power Linnik zones, and exponential inequalities for $\mathbf{P}(\tilde{Z}_{t}\geq x)$ .