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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 the Slepian Gaussian fields
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by Fuchang Gao and Wenbo V. Li PDF
Trans. Amer. Math. Soc. 359 (2007), 1339-1350 Request permission

Abstract:

The $d$-dimensional Slepian Gaussian random field $\{S({\mathbf {t}}), {\mathbf {t}} \in \mathbb {R}_+^d\}$ is a mean zero Gaussian process with covariance function $\mathbb {E} S({\mathbf {s}})S({\mathbf {t}})= \prod _{i=1}^d \max (0, a_i-\left | s_i-t_i\right | )$ for $a_i>0$ and ${\mathbf {t}}=(t_1, \cdots , t_d) \in \mathbb {R}_+^d$. Small ball probabilities for $S({\mathbf {t}})$ are obtained under the $L_2$-norm on $[0,1]^d$, and under the sup-norm on $[0,1]^2$ which implies Talagrand’s result for the Brownian sheet. The method of proof for the sup-norm case is purely probabilistic and analytic, and thus avoids ingenious combinatoric arguments of using decreasing mathematical induction. In particular, Riesz product techniques are new ingredients in our arguments.
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Additional Information
  • Fuchang Gao
  • Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83844
  • MR Author ID: 290983
  • Email: fuchang@uidaho.edu
  • Wenbo V. Li
  • Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
  • Email: wli@math.udel.edu
  • Received by editor(s): October 28, 2004
  • Received by editor(s) in revised form: February 2, 2005
  • Published electronically: October 16, 2006
  • Additional Notes: The first author was supported in part by NSF Grants EPS-0132626 and DMS-0405855
    The second author was supported in part by NSF Grant DMS-0204513
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 1339-1350
  • MSC (2000): Primary 60G15; Secondary 42A55
  • DOI: https://doi.org/10.1090/S0002-9947-06-03963-8
  • MathSciNet review: 2262853