Strong Limit Theorems for Large and Small Increments of $l^p$-Valued Gaussian Processes

Abstract

Based on the well-known Borell inequality and on a general theorem for large and small increments of Banach space valued stochastic processes of Csaki, Csorgo and Shao, we establish some almost sure path behaviour of increments in general, and moduli of continuity in particular, for $l^p$-valued, $1 \leq p < \infty$, Gaussian processes with stationary increments. Applications to $l^p$-valued fractional Wiener and Ornstein-Uhlenbeck processes are also discussed. Our results refine and extend those of Csaki, Csorgo and Shao.