## The Annals of Probability

### 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).

#### Article information

Source
Ann. Probab., Volume 3, Number 4 (1975), 716-721.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996311

Digital Object Identifier
doi:10.1214/aop/1176996311

Mathematical Reviews number (MathSciNet)
MR378070

Zentralblatt MATH identifier
0318.60031

JSTOR