Small ball probabilities for the Slepian Gaussian fields
Authors:
Fuchang Gao and Wenbo V. Li
Journal:
Trans. Amer. Math. Soc. 359 (2007), 13391350
MSC (2000):
Primary 60G15; Secondary 42A55
Published electronically:
October 16, 2006
MathSciNet review:
2262853
Fulltext PDF Free Access
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Abstract: The dimensional Slepian Gaussian random field is a mean zero Gaussian process with covariance function for and . Small ball probabilities for are obtained under the norm on , and under the supnorm on which implies Talagrand's result for the Brownian sheet. The method of proof for the supnorm 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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 Gao, F., Hannig, J., Lee, T.Y. and Torcaso, F. (2003). Laplace transforms via Hadamard factorization, Electron. J. Probab., 8, no. 13, 120. MR 1998764 (2005h:60110)
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 Gao, F. and Li, W.V. (2004). Logarithmic level comparison for small deviation probabilities, J. Theory Probab., 2006, DOI10.1007/s1095900600261 (online).
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 A. Karol, A. Nazarov and Ya. Nikitin (2003), Tensor products of compact operators and logarithmic small ball asymptotics for Gaussian random fields, Preprint.
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 Kuelbs, J. and Li, W.V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116, 133157.MR 1237989 (94j:60078)
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 Li, W.V. (1992). Comparison results for the lower tail of Gaussian seminorms, J. Theoret. Probab, 5, 131. MR 1144725 (93k:60088)
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 Li, W.V. and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27, 15561578. MR 1733160 (2001c:60059)
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 Li, W.V. and Shao, Q.M. (2001). Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods, Handbook of Statist., Vol. 19, 533597. MR 1861734
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 Lifshits, M.A., and Tsyrelson, B.S. (1986). Small ball deviations of Gaussian fields. Theor. Probab. Appl. 31, 557558.
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 Martin, A. (2004). Small ball asymptotics for the stochastic wave equation. J. Theoret. Probab. 17, 693703. MR 2091556 (2005h:60188)
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 Temlyakov, V. (1995). An inequality for trigonometric polynomials and its application for estimating the entropy numbers. Journal of Complexity 11, 293307.MR 1334238 (96c:41052)
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Additional Information
Fuchang Gao
Affiliation:
Department of Mathematics, University of Idaho, Moscow, Idaho 83844
Email:
fuchang@uidaho.edu
Wenbo V. Li
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email:
wli@math.udel.edu
DOI:
http://dx.doi.org/10.1090/S0002994706039638
PII:
S 00029947(06)039638
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 EPS0132626 and DMS0405855
The second author was supported in part by NSF Grant DMS0204513
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
