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# Tail Behaviour for the Suprema of Gaussian Processes with Applications to Empirical Processes

The Annals of Probability
Vol. 15, No. 4 (Oct., 1987), pp. 1339-1351
Stable URL: http://www.jstor.org/stable/2244006
Page Count: 13
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## Abstract

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 λ large. Under the assumption that the entropy function of C grows polynomially, we find bounds of the form cλαexp(- 1/2λ2/σ2), where σ2 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 α is identified. These include the distributions of the maxima of the rectangle indexed, pinned Brownian sheet on Rk for which α = 2(2k - 1), and the half plane indexed pinned sheet on R2 for which α = 2.

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