Spectral criteria, SLLN's and A.S. convergence of series of stationary variables

Abstract

It is shown here how to extend the spectral characterization of the strong law of large numbers for weakly stationary processes to certain singular averages. For instance, letting $\{X_t, t \in R^3\}$ be a weakly stationary field, $\{X_t\}$ satisfies the usual SLLN (by averaging over balls) if and only if the averages of $\{X_t\}$ over spheres of increasing radii converge pointwise. The same result in two dimensions is false. This spectral approach also provides a necessary and sufficient condition for the a.s. convergence of some series of stationary variables.