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.
Primary Subjects: 60F15, 60G10, 60G60
Secondary Subjects: 60G12, 47A35
References
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