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

Peter Kern (), Mark M. Meerschaert () and Yimin Xiao ()
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Peter Kern: Heinrich-Heine-Universität Düsseldorf
Mark M. Meerschaert: Michigan State University
Yimin Xiao: Michigan State University

Journal of Theoretical Probability, 2018, vol. 31, issue 1, 598-617

Abstract: 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.

Keywords: Lévy exponent; Operator semistable process; Semi-selfsimilarity; Hausdorff dimension; Packing dimension; Range; Graph; Multiple points; Recurrence; Transience; Primary 60E10; 60G51; Secondary 28A78; 28A80; 60E07; 60G17; 60G18; 60G52 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-016-0720-6

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