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April, 1986 Sample Moduli for Set-Indexed Gaussian Processes
Kenneth S. Alexander
Ann. Probab. 14(2): 598-611 (April, 1986). DOI: 10.1214/aop/1176992533


Sample path behavior is studied for Gaussian processes $W_p$ indexed by classes $\mathscr{L}$ of subsets of a probability space $(X, \mathscr{A}, P)$ with covariance $EW_P(A)W_P(B) = P(A \cap B)$. A function $\psi$ is found in some cases such that $\lim \sup_{t\rightarrow 0}\sup\{|W_P(C)|/\psi(P(C)): C \in \mathscr{L}, P(C) \leq t\} = 1$ a.s. This unifies and generalizes the LIL and Levy's Holder condition for Brownian motion, and some results of Orey and Pruitt for the Brownian sheet.


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Kenneth S. Alexander. "Sample Moduli for Set-Indexed Gaussian Processes." Ann. Probab. 14 (2) 598 - 611, April, 1986.


Published: April, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0601.60037
MathSciNet: MR832026
Digital Object Identifier: 10.1214/aop/1176992533

Primary: 60G15
Secondary: 60G17

Keywords: Gaussian process , sample modulus , set-indexed process , Vapnik-Cervonenkis class

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 2 • April, 1986
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