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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Small ball probabilities for Gaussian random fields and tensor products of compact operators
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by Andrei Karol’, Alexander Nazarov and Yakov Nikitin PDF
Trans. Amer. Math. Soc. 360 (2008), 1443-1474 Request permission

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
  • 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.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2357702