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

In this paper, we consider a random variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Z_{t}=\sum_{i=1}^{N_{t}}a_{i}X_{i}$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X, X_{1}, X_{2}, \ldots$\end{document} 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 integer-valued 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{Z}_{t}=(\mathbf{D}Z_{t})^{-1/2}(Z_{t}-\mathbf{E}Z_{t})$\end{document}, large deviation theorems in the Cramer and power Linnik zones, and exponential inequalities for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{P}(\tilde{Z}_{t}\geq x)$\end{document}.