Abstract: We establish a maximal moment inequality for the weighted sum of a sequence of random variables with finite second moments. An extension to the Hájek-Rény and Chow inequalities is then obtained. When certain second-moment properties are fulfilled, it enables us to deduce a strong law for the weighted sum of a time series having long-range dependence. Applications to estimation and model selection in multiple regression models with long-range dependent errors are also given.
Key words and phrases: Autoregressive fractionally integrated moving average, convergence system, long-range dependence, maximal inequality, model selection, strong consistency.