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Xiao, Yimin

Hausdorff-type measures of the sample paths of fractional Brownian motion. (English) Zbl 0937.60076

Stochastic Processes Appl. 74, No. 2, 251-272 (1998).
Summary: Let \(\varphi \) be a Hausdorff measure function and let \(\Lambda \) be an infinite increasing sequence of positive integers. The Hausdorff-type measure \(\varphi -m_{\Lambda}\) associated to \(\varphi \) and \(\Lambda \) is studied. Let \(X(t)\) \((t\in \mathbb R^N)\) be fractional Brownian motion of index \(\alpha \) in \(\mathbb R^d\). We evaluate the exact \(\varphi -m_{\Lambda}\) measure of the image and graph set of \(X(t)\). A necessary and sufficient condition on the sequence \(\Lambda \) is given so that the usual Hausdorff measure functions for \(X ([0,1]^N)\) and \(\text{Gr }X([0,1]^N)\) are still the correct measure functions. If the sequence \(\Lambda \) increases faster, then some smaller measure functions will give positive and finite \((\varphi ,\Lambda)\)-Hausdorff measure for \(X([0,1]^N)\) and \(\text{Gr } X([0,1]^N)\).

Cited in 6 Documents

MSC:

60J65 Brownian motion
28A78 Hausdorff and packing measures
60G17 Sample path properties
28A80 Fractals

Keywords:

fractional Brownian motion; image; graph set; Hausdorff-type measures; packing measure
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References:

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