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Multiple Points of Operator Semistable Lévy Processes

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  • Tomasz Luks

    (Universität Paderborn)

  • Yimin Xiao

    (Michigan State University)

Abstract

We determine the Hausdorff dimension of the set of k-multiple points for a symmetric operator semistable Lévy process $$X=\{X(t), t\in {\mathbb {R}}_+\}$$X={X(t),t∈R+} in terms of the eigenvalues of its stability exponent. We also give a necessary and sufficient condition for the existence of k-multiple points. Our results extend to all $$k\ge 2$$k≥2 the recent work (Luks and Xiao in J Theor Probab 30(1):297–325, 2017) where the set of double points $$(k = 2)$$(k=2) was studied in the symmetric operator stable case.

Suggested Citation

  • Tomasz Luks & Yimin Xiao, 2020. "Multiple Points of Operator Semistable Lévy Processes," Journal of Theoretical Probability, Springer, vol. 33(1), pages 153-179, March.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:1:d:10.1007_s10959-018-0859-4
    DOI: 10.1007/s10959-018-0859-4
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    References listed on IDEAS

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    1. Laha, R. G. & Rohatgi, V. K., 1980. "Semistable measures on a Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 10(1), pages 88-94, March.
    2. Makoto Maejima & Ken-iti Sato, 1999. "Semi-Selfsimilar Processes," Journal of Theoretical Probability, Springer, vol. 12(2), pages 347-373, April.
    3. 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.
    4. 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.
    5. 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.
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