Central limit theorems for moving average processes*

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $$\end{document} be a stationary sequence of real random variables with Eξ0 = 0 and infinite variance. Furthermore, assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {{\left( {{c_n}} \right)}_{{n\in \mathbb{Z}}}} $$\end{document} is a sequence of real numbers and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {X_n}=\sum {_{{j\in \mathbb{Z}}}{c_j}{\xi_{n-j }}} $$\end{document} is a moving average processes driven by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $$\end{document}. By using a decomposition of the moving average processes, a central limit theorem for the partial sums \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sum\nolimits_{k=1}^n {{X_k}} $$\end{document} is established. As applications, we obtain some central limit theorems for stationary dependent sequences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $$\end{document}, such as associated sequence, martingale difference, and so on.