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Tail behaviour for the suprema of Gaussian processes with applications to empirical processes. (English) Zbl 0638.60059

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 [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?
Reviewer: E.Gine

MSC:

60G15 Gaussian processes
60G57 Random measures
60F10 Large deviations
62G30 Order statistics; empirical distribution functions
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