Some Limit Theorems for Delayed Sums of Dependent Random Sequence

In this paper we introduce a new concept of asymptotic logarithmic delayed likelihood ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{LR}(w)}$$\end{document} as a measure of deviation of the joint distribution from the product of their margins. In the continuous case, we prove the generalized strong law of large numbers for delayed averages ρn, kn, i.e. strong deviation theorems (strong law expressed by inequalities).