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April, 1995 Hausdorff Measure of Trajectories of Multiparameter Fractional Brownian Motion
Michel Talagrand
Ann. Probab. 23(2): 767-775 (April, 1995). DOI: 10.1214/aop/1176988288


Consider $0 < \alpha < 1$ and the Gaussian process $Y(t)$ on $\mathbb{R}^N$ with covariance $E(Y(t)Y(s)) = |t|^{2\alpha} + |s|^{2\alpha} - |t - s|^{2\alpha}$, where $|t|$ is the Euclidean norm of $t$. Consider independent copies $X^1,\ldots,X^d$ of $Y$ and the process $X(t) = (X^1(t),\ldots,X^d(t))$ valued in $\mathbb{R}^d$. In the transient case $(N < \alpha d)$ we show that a.s. for each compact set $L$ of $\mathbb{R}^N$ with nonempty interior, we have $0 < \mu_\varphi(X(L)) < \infty$, where $\mu_\varphi$ denotes the Hausdorff measure associated with the function $\varphi(\varepsilon) = \varepsilon^{N/\alpha} \log \log(1/\varepsilon)$. This result extends work of A. Goldman in the case $\alpha = 1/2$; the proofs are considerably simpler.


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Michel Talagrand. "Hausdorff Measure of Trajectories of Multiparameter Fractional Brownian Motion." Ann. Probab. 23 (2) 767 - 775, April, 1995.


Published: April, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0830.60034
MathSciNet: MR1334170
Digital Object Identifier: 10.1214/aop/1176988288

Primary: 60G15
Secondary: 26B15 , 60G17

Keywords: Brownian motion , Haussdorff dimension

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 2 • April, 1995
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