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

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Abstract

This paper proves sharp bounds on the tails of the Lévy exponent of an operator semistable law on \({\mathbb R^d}\). 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.

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Authors and Affiliations

  1. Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstr 1, 40225, Düsseldorf, Germany

    Peter Kern

  2. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Mark M. Meerschaert & Yimin Xiao

Authors
  1. Peter Kern
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  2. Mark M. Meerschaert
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  3. Yimin Xiao
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Corresponding author

Correspondence to Peter Kern.

Additional information

The work of Peter Kern was supported by Deutsche Forschungsgemeinschaft (DFG) under Grant KE1741/6-1. Mark M. Meerschaert was partially supported by ARO Grant W911NF-15-1-0562 and NSF Grants DMS-1462156 and EAR-1344280. Yimin Xiao was partially supported by NSF Grants DMS-1307470, DMS-1309856 and DMS-1612885.

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Kern, P., Meerschaert, M.M. & Xiao, Y. Asymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path Properties. J Theor Probab 31, 598–617 (2018). https://doi.org/10.1007/s10959-016-0720-6

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  • DOI: https://doi.org/10.1007/s10959-016-0720-6

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