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Small ball probabilities for Gaussian random fields and tensor products of compact operators

Authors: Andrei Karol', Alexander Nazarov and Yakov Nikitin
Journal: Trans. Amer. Math. Soc. 360 (2008), 1443-1474
MSC (2000): Primary 60G15; Secondary 60G60, 47A80
Published electronically: October 23, 2007
MathSciNet review: 2357702
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Abstract: We find the logarithmic $ L_2$-small ball asymptotics for a large class of zero mean Gaussian fields with covariances having the structure of ``tensor product''. The main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of Gaussian random functions including the fractional Brownian sheet, Ornstein - Uhlenbeck sheet, etc. So we get the far-reaching generalizations of well-known results by Csáki (1982) and by Li (1992). Another class of Gaussian fields considered is the class of additive fields studied under the supremum-norm by Chen and Li (2003). Our theorems are based on new results on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which are of independent interest.

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Additional Information

Andrei Karol'
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr., 28, St. Petersburg, 198504, Russia

Alexander Nazarov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr., 28, St. Petersburg, 198504, Russia

Yakov Nikitin
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr., 28, St. Petersburg, 198504, Russia

Keywords: Small deviations, fractional Brownian motion, Brownian sheet, Ornstein -- Uhlenbeck sheet, tensor product of operators, spectral asymptotics, slowly varying functions.
Received by editor(s): April 24, 2005
Received by editor(s) in revised form: November 22, 2005
Published electronically: October 23, 2007
Additional Notes: The authors were partially supported by RFBR Grant 04-01-00716.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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