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October, 1987 Tail Behaviour for the Suprema of Gaussian Processes with Applications to Empirical Processes
Robert J. Adler, Gennady Samorodnitsky
Ann. Probab. 15(4): 1339-1351 (October, 1987). DOI: 10.1214/aop/1176991980


Initially we consider "the" standard isonormal linear process $L$ on a Hilbert space $H$, and applying metric entropy methods obtain bounds for the probability that $\sup_CLx > \lambda, C \subset H$ and $\lambda$ large. Under the assumption that the entropy function of $C$ grows polynomially, we find bounds of the form $c\lambda^\alpha\exp(- \frac{1}{2}\lambda^2/\sigma^2)$, where $\sigma^2$ is the maximal variance of $L$. We use a notion of entropy finer than that usually employed and specifically suited to the nonstationary situation. As a result we obtain, in the nonstationary setting, more precise bounds than any in the literature. We then treat a number of examples in which the power $\alpha$ is identified. These include the distributions of the maxima of the rectangle indexed, pinned Brownian sheet on $\mathbb{R}^k$ for which $\alpha = 2(2k - 1)$, and the half plane indexed pinned sheet on $\mathbb{R}^2$ for which $\alpha = 2$.


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Robert J. Adler. Gennady Samorodnitsky. "Tail Behaviour for the Suprema of Gaussian Processes with Applications to Empirical Processes." Ann. Probab. 15 (4) 1339 - 1351, October, 1987.


Published: October, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0638.60059
MathSciNet: MR905335
Digital Object Identifier: 10.1214/aop/1176991980

Primary: 60G15
Secondary: 60F10 , 60G57 , 62G30

Keywords: Brownian sheet , Empirical processes , Gaussian processes , isonormal process , Metric entropy , supremum

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 4 • October, 1987
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