The Functional Law of the Iterated Logarithm for the Empirical Process Based on Sample Means

Consider a double array \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ {X_{i,j} ;i \geqslant 1,j \geqslant } \right\}$$ \end{document} of i.i.d. random variables with mean μ and variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sigma ^2 (0 < \sigma ^2 < \infty )$$ \end{document} and set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Z_{i,n} = n^{ - 1/2} \sum\nolimits_{j = 1}^n {(X_{i,j} - \mu )} /\sigma $$ \end{document}. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat \Phi _{N,n} $$ \end{document} denote the empirical distribution function of Z1, n,..., ZN, n and let Φ be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sqrt N (\hat \Phi _{N,n} - \Phi )$$ \end{document}, where n=n(N)→∞ as N→∞. For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented.