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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Packing measure of the sample paths
of fractional Brownian motion


Author: Yimin Xiao
Journal: Trans. Amer. Math. Soc. 348 (1996), 3193-3213
MSC (1991): Primary 60G15, 60G17
MathSciNet review: 1370655
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Abstract: Let $X(t) \ (t \in % \mathbf {R}) $ be a fractional Brownian motion of index $% \alpha $ in $% \mathbf {R}^d.$ If $1 < % \alpha d\ $, then there exists a positive finite constant $K $ such that with probability 1,

\begin{displaymath}\hbox { $\phi $-$p(X([0,t]))$} = Kt \ \ \ \hbox {for any } \ t > 0\ ,\end{displaymath}

where $% \phi (s) = s^{\frac 1 {% \alpha }}/ (\log \log \frac 1 s)^{\frac 1 {2 % \alpha }}$ and $\phi $-$p (X([0,t]))$ is the $\phi $-packing measure of $X([0,t])$.


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Additional Information

Yimin Xiao
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Address at time of publication: Department of Mathematic, University of Utah, Salt Lake City, Utah 84112

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01712-6
PII: S 0002-9947(96)01712-6
Keywords: Packing measure, fractional Brownian motion, image, sojourn time
Received by editor(s): August 2, 1995
Article copyright: © Copyright 1996 American Mathematical Society