On the Almost Sure Convergence of Series of Stationary and Related Nonstationary Variables

Abstract

Let $\{X_n\}$ be, for example, a weakly stationary sequence or a lacunary system with finite $p$th moment, $1 \leq p \leq 2$, and let $\{a_n\}$ be a sequence of scalars. We obtain here conditions which ensure the almost sure convergence of the series $\sum a_nX_n$. When $\{X_n\}$ is an orthonormal sequence, the classical Rademacher-Menchov theorem is recovered. This is then applied to study the strong consistency of least squares estimates in multiple regression models.