(Georgia Institute of Technology, 1995-04)
Houdré, Christian; Lacey, Michael T.
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 provide a necessary and sufficient
condition for the a.s. convergence of some series of stationary variables.