On the rate of convergence in the global central limit theorem for random sums of independent random variables

We present upper bounds of the integral ∫−∞∞xlPZN<x−Φxdx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathbf{P}\left\{{Z}_N<x\right\}-\varPhi (x)\right|\mathrm{d}x $$\end{document} for 0 ≤ l ≤ 1 + δ, where 0 < δ ≤ 1, Φ(x) is a standard normal distribution function, and ZN = SN/VSN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {S}_N/\sqrt{\mathbf{V}{S}_N} $$\end{document} is the normalized random sum with variance VSN > 0 (SN = X1 + · · · + XN) of centered independent random variables X1,X2, . . . . The number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .