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April 1996 Spectral criteria, SLLN's and A.S. convergence of series of stationary variables
C. Houdré, M. T. Lacey
Ann. Probab. 24(2): 838-856 (April 1996). DOI: 10.1214/aop/1039639364


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.


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C. Houdré. M. T. Lacey. "Spectral criteria, SLLN's and A.S. convergence of series of stationary variables." Ann. Probab. 24 (2) 838 - 856, April 1996.


Published: April 1996
First available in Project Euclid: 11 December 2002

zbMATH: 0868.60025
MathSciNet: MR1404530
Digital Object Identifier: 10.1214/aop/1039639364

Primary: 60F15 , 60G10 , 60G60
Secondary: 47A35 , 60G12

Keywords: a.s. convergence , Calderón-Zygmund kernel , homogeneous field , Spherical means , stationary process , unitary group

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 2 • April 1996
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