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Multiple Points of Operator Semistable Lévy Processes

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Abstract

We determine the Hausdorff dimension of the set of k-multiple points for a symmetric operator semistable Lévy process \(X=\{X(t), t\in {\mathbb {R}}_+\}\) 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 \(k\ge 2\) the recent work (Luks and Xiao in J Theor Probab 30(1):297–325, 2017) where the set of double points \((k = 2)\) was studied in the symmetric operator stable case.

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Author information

Correspondence to Tomasz Luks.

Additional information

Research of Y. Xiao was partially supported by the NSF Grants DMS-1612885 and DMS-1607089.

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Cite this article

Luks, T., Xiao, Y. Multiple Points of Operator Semistable Lévy Processes. J Theor Probab 33, 153–179 (2020). https://doi.org/10.1007/s10959-018-0859-4

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Keywords

  • Multiple points
  • Hausdorff dimension
  • Operator semistable process
  • Lévy process

Mathematics Subject Classification (2010)

  • 60J25
  • 60J30
  • 60G51
  • 60G17