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Lacey,
Michael T.
Lacey,
Michael T.
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ItemSpectral, Criteria, SLLNS and A.S. Convergence of Series of Stationary Variables(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.