Inequalities for Increments of Stochastic Processes and Moduli of Continuity

Abstract

Let $\{\Gamma(t), t \in \mathbb{R}\}$ be a Banach space $\mathscr{B}$-valued stochastic process. Let $P$ be the probability measure generated by $\Gamma(\cdot)$. Assume that $\Gamma(\cdot)$ is $P$-almost surely continuous with respect to the norm $\| \|$ of $\mathscr{B}$ and that there exists a positive nondecreasing function $\sigma(a), a > 0$, such that $P\{\|\Gamma(t + a) - \Gamma(t)\| \geq x\sigma(a)\} \leq K \exp(-\gamma x^\beta)$ with some $K, \gamma, \beta > 0$. Then, assuming also that $\sigma(\cdot)$ is a regularly varying function at zero, or at infinity, with a positive exponent, we prove large deviation results for increments like $\sup_{0\leq t\leq T-a}\sup_{0\leq s\leq a}\|\Gamma(t + s) - \Gamma(t)\|$, which we then use to establish moduli of continuity and large increment estimates for $\Gamma(\cdot)$. One of the many applications is to prove moduli of continuity estimates for $l^2$-valued Ornstein-Uhlenbeck processes.