Tail behaviour for the suprema of Gaussian processes with applications to empirical processes
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?
probability bound for the supremum of a Gaussian process
scientific article; zbMATH DE number 4040968