## The Annals of Probability

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

### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60G15: Gaussian processes

Secondary: 60G17: Sample path properties

**Keywords**

Gaussian processes quadratic variation

#### Citation

Klein, Ruben; Gine, Evarist. On Quadratic Variation of Processes with Gaussian Increments. Ann. Probab. 3 (1975), no. 4, 716--721. doi:10.1214/aop/1176996311. https://projecteuclid.org/euclid.aop/1176996311