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Asymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path Properties

Author

Listed:
  • Peter Kern

    (Heinrich-Heine-Universität Düsseldorf)

  • Mark M. Meerschaert

    (Michigan State University)

  • Yimin Xiao

    (Michigan State University)

Abstract

This paper proves sharp bounds on the tails of the Lévy exponent of an operator semistable law on $${\mathbb R^d}$$ R d . These bounds are then applied to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-selfsimilar Lévy processes. The proofs are elementary, using only the properties of the Lévy exponent, and certain index formulae.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:1:d:10.1007_s10959-016-0720-6
    DOI: 10.1007/s10959-016-0720-6
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    References listed on IDEAS

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    1. Makoto Maejima & Ken-iti Sato, 1999. "Semi-Selfsimilar Processes," Journal of Theoretical Probability, Springer, vol. 12(2), pages 347-373, April.
    2. 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.
    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. 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.
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    Cited by:

    1. Peter Kern & Svenja Lage, 2023. "On Self-Similar Bernstein Functions and Corresponding Generalized Fractional Derivatives," Journal of Theoretical Probability, Springer, vol. 36(1), pages 348-371, March.
    2. Tomasz Luks & Yimin Xiao, 2020. "Multiple Points of Operator Semistable Lévy Processes," Journal of Theoretical Probability, Springer, vol. 33(1), pages 153-179, March.

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