Packing measure of the sample paths of fractional Brownian motion

Xiao, Yimin

doi:10.1090/S0002-9947-96-01712-6

Packing measure of the sample paths of fractional Brownian motion
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by Yimin Xiao PDF

Trans. Amer. Math. Soc. 348 (1996), 3193-3213 Request permission

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, \[ \phi -p(X([0,t])) = Kt\quad \text {for any $t > 0$}, \] 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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