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