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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-07-04233-X
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
Email: karol@ak1078.spb.edu

Alexander Nazarov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr., 28, St. Petersburg, 198504, Russia
MR Author ID: 228194
Email: an@AN4751.spb.edu

Yakov Nikitin
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr., 28, St. Petersburg, 198504, Russia
Email: yanikit47@mail.ru

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.