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Dimension results for sample paths of operator stable Lévy processes

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  • Meerschaert, Mark M.
  • Xiao, Yimin

Abstract

Let X= X(t),t[set membership, variant]R+ be an operator stable Lévy process in Rd with exponent B, where B is an invertible linear operator on Rd. We determine the Hausdorff dimension and the packing dimension of the range X([0,1]) in terms of the real parts of the eigenvalues of B.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:1:p:55-75
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    References listed on IDEAS

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    1. Xiao, Yimin, 1997. "Packing dimension of the image of fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 379-387, May.
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    Cited by:

    1. Tomasz Luks & Yimin Xiao, 2020. "Multiple Points of Operator Semistable Lévy Processes," Journal of Theoretical Probability, Springer, vol. 33(1), pages 153-179, March.
    2. Cohen, Serge & Meerschaert, Mark M. & Rosinski, Jan, 2010. "Modeling and simulation with operator scaling," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2390-2411, December.
    3. Tomasz Luks & Yimin Xiao, 2017. "On the Double Points of Operator Stable Lévy Processes," Journal of Theoretical Probability, Springer, vol. 30(1), pages 297-325, March.
    4. Peter Kern & Lina Wedrich, 2014. "The Hausdorff Dimension of Operator Semistable Lévy Processes," Journal of Theoretical Probability, Springer, vol. 27(2), pages 383-403, June.
    5. Lőrinczi, József & Yang, Xiaochuan, 2019. "Multifractal properties of sample paths of ground state-transformed jump processes," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 83-94.
    6. Peter Kern & Mark M. Meerschaert & Yimin Xiao, 2018. "Asymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path Properties," Journal of Theoretical Probability, Springer, vol. 31(1), pages 598-617, March.
    7. Hou, Yanyan & Ying, Jiangang & Dai, Chaoshou, 2008. "Fractal sets determined by dilation-stable processes," Chaos, Solitons & Fractals, Elsevier, vol. 38(3), pages 852-863.
    8. 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.

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