ON THE ALMOST SURE CONVERGENCE OF SERIES OF STATIONARY AND RELATED NONSTATIONARY VARIABLES

Let (X(n)) be, for example, a weakly stationary sequence or a lacunary system with finite pth moment, 1 less than or equal to p less than or equal to 2, and let {a(n)} be a sequence of scalars. We obtain here c onditions which ensure the almost sure convergence of the series Sigma a(n)X(n). When {X(n)} is an orthonormal sequence, the classical Radem acher-Menchov theorem is recovered. This is then applied to study the strong consistency of least squares estimates in multiple regression m odels.