Sample moduli for set-indexed Gaussian processes. (English) Zbl 0601.60037

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 S. Orey and W. E. Pruitt’s [ibid. 1, 138-163 (1973; Zbl 0284.60036)] statements about Brownian sheet (Bs). But note that precise results of 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.
Reviewer: P.Nobelis


60G15 Gaussian processes
60G17 Sample path properties
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