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Hausdorff-type measures of the sample paths of fractional Brownian motion

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  • Xiao, Yimin

Abstract

Let [phi] be a Hausdorff measure function and let [Lambda] be an infinite increasing sequence of positive integers. The Hausdorff-type measure [phi]-m[Lambda] associated to [phi] and [Lambda] is studied. Let be fractional Brownian motion of index [alpha] in We evaluate the exact [phi]-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 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 ([phi],[Lambda])-Hausdorff measure for X([0,1]N) and Gr X([0,1]N)

Suggested Citation

  • Xiao, Yimin, 1998. "Hausdorff-type measures of the sample paths of fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 74(2), pages 251-272, June.
    • Handle: RePEc:eee:spapps:v:74:y:1998:i:2:p:251-272
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    Cited by:

    1. Dhifaoui, Zouhaier & Bardet, Jean-Marc, 2022. "Local correlation dimension of multidimensional stochastic process," Statistics & Probability Letters, Elsevier, vol. 181(C).

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