On Quadratic Variation of Processes with Gaussian Increments

Abstract

This note extends to a broad class of stochastic processes with Gaussian increments the following theorem of R. M. Dudley (Ann. Probability 1 66-103): if $\{\pi_n\}$ is any sequence of partitions of [0, 1] with mesh $(\pi_n) = o(1/\log n)$ and if $L(\pi_n)^2$ is the quadratic variation of Brownian motion corresponding to $\pi_n$, then a.s.-$\lim_{n\rightarrow\infty}L(\pi_n)^2 = 1$. (Actually, Dudley proves a more general theorem). The main tool used is a bound of exponential type for the tail probabilities of quadratic functions of Gaussian random variables (Hanson and Wright, Ann. Math. Statist. 42 1079-1083).