Sample Moduli for Set-Indexed Gaussian Processes

Abstract

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.