Title:
Spectral, Criteria, SLLNS and A.S. Convergence of Series of Stationary Variables
Spectral, Criteria, SLLNS and A.S. Convergence of Series of Stationary Variables
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Author(s)
Houdré, Christian
Lacey, Michael T.
Lacey, Michael T.
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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 provide a necessary and sufficient
condition for the a.s. convergence of some series of stationary variables.
Sponsor
Research of both authors supported in part by NSF Postdoctoral Fellowships. The first named author was also supported in part a NSF-NATO Postdoctoral fellowship, while at CEREMADE,
Universite Paris{Dauphine, 75775 Paris Cedex, France and at CERMA, ENPC, La Courtine, 93167
Noisy le Grand Cedex, France.
Date Issued
1995-04
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Text
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Pre-print