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August, 1975 On Quadratic Variation of Processes with Gaussian Increments
Ruben Klein, Evarist Gine
Ann. Probab. 3(4): 716-721 (August, 1975). DOI: 10.1214/aop/1176996311


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).


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Ruben Klein. Evarist Gine. "On Quadratic Variation of Processes with Gaussian Increments." Ann. Probab. 3 (4) 716 - 721, August, 1975.


Published: August, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0318.60031
MathSciNet: MR378070
Digital Object Identifier: 10.1214/aop/1176996311

Primary: 60G15
Secondary: 60G17

Keywords: Gaussian processes , Quadratic Variation

Rights: Copyright © 1975 Institute of Mathematical Statistics

Vol.3 • No. 4 • August, 1975
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