A maximal moment inequality for long range dependent time series with applications to estimation and model selection

We establish a maximal moment inequality for the weighted sum of a long- range dependent process. An extension to H$\acute{a}$jek-R$\acute{e}$ny and Chow's type inequality is then obtained. It enables us to deduce a strong law for the weighted sum of a stationary long-range dependent time series. To illustrate its usefulness, applications of the inequality to estimation and model selection in multiple regression models with long-range dependent errors are given.