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Small ball probabilities for the Slepian Gaussian fields

Authors: Fuchang Gao and Wenbo V. Li
Journal: Trans. Amer. Math. Soc. 359 (2007), 1339-1350
MSC (2000): Primary 60G15; Secondary 42A55
Published electronically: October 16, 2006
MathSciNet review: 2262853
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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\vert s_i-t_i\right\vert )$ 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

Wenbo V. Li
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716

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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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