Sample moduli for set-indexed Gaussian processes
type: Item
instance of: scholarly article
title: Sample moduli for set-indexed Gaussian processes
zbMATH Open document ID: 0601.60037
author: Kenneth S. Alexander
published in: The Annals of Probability
publication date: 1986-01-01T00:00:00Z
review text: The author studies sample path moduli for Gaussian processes W indexed by Vapnik-Červonenkis classes \({\mathcal C}\) of subsets of a probability space (X,\({\mathcal A},P)\) with covariance E W(A)W(B)\(=P(A\cap B)\). The main result of this paper is that in some cases there exist a function \(\psi\) such that: \[ \sup_{t\to 0}\sup \{| W(C)| /\psi (P(C)): C\in {\mathcal C},\quad P(C)\leq t\}=1\quad a.s. \] The function \(\psi\) is given explicitly and depends on the capacity function, introduced by the author, and the index of \({\mathcal C}\) for P. This theorem unifies a number of results on sample moduli for usual Gaussian processes: the standard LIL, Levy's Hölder condition for the Brownian motion (Bm), and \textit{S. Orey} and \textit{W. E. Pruitt}'s [ibid. 1, 138-163 (1973; Zbl 0284.60036)] statements about Brownian sheet (Bs). But note that precise results of \textit{C. Mueller} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 163-179 (1981; Zbl 0463.60031)] on the increments of Bm and results of Orey and Pruitt on the increments of Bs, established both of them by using the Markov properties of such processes, can not be obtained with the present general theorem.
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 3969776
zbMATH Keywords: set-indexed process
zbMATH Keywords: sample path moduli for Gaussian processes
zbMATH Keywords: Vapnik- Červonenkis classes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992533
OpenAlex ID: W2077347126
DOI: 10.1214/AOP/1176992533
description: scientific article; zbMATH DE number 3969776