Tail behaviour for the suprema of Gaussian processes with applications to empirical processes
type: Item
instance of: scholarly article
title: Tail behaviour for the suprema of Gaussian processes with applications to empirical processes
zbMATH Open document ID: 0638.60059
author: Robert J. Adler
author: Gennady Samorodnitsky
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: A tail probability bound for the supremum of a Gaussian process is obtained by localization of Fernique's bound in the case of metric entropy of polynomial type. This is then used to give sharp upper bounds of the form \(\lambda^{\alpha} \exp (-\lambda^ 2/\sigma^ 2)\), \(\lambda\) large, for the tail probability of the sup of the set indexed Brownian sheet and the pinned Brownian sheet (which appears as the limit to the empirical process indexed by sets). Reviewer's remark: replacing entropy by a majorizing measure (m.m.) is also a localization procedure, which actually gives two-sided bounds [\textit{M. Talagrand}, Regularity of Gaussian processes, Acta Math. 159, 99-149 (1987)]. Of course there is always the problem of finding the adequate m.m. Could the result in this article be obtained (perhaps in an improved and/or extended form) in the framework of m.m.? Could the sharp bounds in the examples be derived directly from the m.m. theorem?
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 4040968
zbMATH Keywords: probability bound for the supremum of a Gaussian process
zbMATH Keywords: metric entropy
zbMATH Keywords: Brownian sheet
zbMATH Keywords: localization procedure
reviewed by: Evarist Giné
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991980
OpenAlex ID: W2001567961
DOI: 10.1214/AOP/1176991980
description: scientific article; zbMATH DE number 4040968