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The Hausdorff dimension of the range of the Lévy multistable processes

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  • R. Guével

    (Univ Rennes, CNRS, IRMAR - UMR 6625)

Abstract

We compute the Hausdorff dimension of the image X(E) of a non-random Borel set $$E \subset [0,1]$$ E ⊂ [ 0 , 1 ] , where X is a Lévy multistable process in $$\mathbf{R}.$$ R . This extends the case where X is a classical stable Lévy process by letting the stability exponent $$\alpha $$ α be a smooth function. Hence, we are considering here non-homogeneous processes with increments which are not stationary and not necessarily independent. Contrary to the situation where the stability parameter is a constant, the dimension depends on the version of the multistable Lévy motion when the process has an infinite first moment.

Suggested Citation

  • R. Guével, 2019. "The Hausdorff dimension of the range of the Lévy multistable processes," Journal of Theoretical Probability, Springer, vol. 32(2), pages 765-780, June.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:2:d:10.1007_s10959-018-0847-8
    DOI: 10.1007/s10959-018-0847-8
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    References listed on IDEAS

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    1. Ayache, Antoine, 2013. "Sharp estimates on the tail behavior of a multistable distribution," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 680-688.
    2. Meerschaert, Mark M. & Xiao, Yimin, 2005. "Dimension results for sample paths of operator stable Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 55-75, January.
    3. K. J. Falconer & J. Lévy Véhel, 2009. "Multifractional, Multistable, and Other Processes with Prescribed Local Form," Journal of Theoretical Probability, Springer, vol. 22(2), pages 375-401, June.
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