On normal approximation of discounted and strongly mixing random variables

We estimate the difference \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left| {\mathbb{E}h(Z_v ) - \mathbb{E}h(N)} \right|$$ \end{document} for bounded functions h: ℝ → ℝ satisfying the Lipschitz condition, where Zv = Bv−1 ∑i=0∞viXi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$B_v^2 = \mathbb{E}(\sum\nolimits_{i = 0}^\infty {\upsilon ^i X_i } )^2 > 0$$ \end{document} with discount factor ν such that 0 < ν < 1. Here {Xn, n ≥ 0} is a sequence of strongly mixing random variables with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{E}X_n = 0$$ \end{document}, and N is a standard normal random variable. In a particular case, the obtained upper bounds are of order O((1 − ν)1/2).