The Hausdorff Dimension of Operator Semistable Lévy Processes
Peter Kern,Lina Wedrich +1 more
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TLDR
Meerschaert and Xiao as discussed by the authors considered the special case of an operator stable (selfsimilar) Levy process and determined the Hausdorff dimension of the partial range X(B) in terms of the real parts of the eigenvalues of E.Abstract:
Let X={X(t)}t≥0 be an operator semistable Levy process in ℝd with exponent E, where E is an invertible linear operator on ℝd and X is semi-selfsimilar with respect to E. By refining arguments given in Meerschaert and Xiao (Stoch. Process. Appl. 115, 55–75, 2005) for the special case of an operator stable (selfsimilar) Levy process, for an arbitrary Borel set B⊆ℝ+ we determine the Hausdorff dimension of the partial range X(B) in terms of the real parts of the eigenvalues of E and the Hausdorff dimension of B.read more
Citations
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Hausdorff dimension of the graph of an operator semistable Lévy process
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Asymptotic behavior of semistable L\'evy exponents and applications to fractal path properties
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Asymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path Properties
TL;DR: In this paper, the authors prove sharp bounds on the tails of the Levy exponent of an operator semistable law on a given graph and apply these bounds 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 Levy processes.
References
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Fractal Geometry: Mathematical Foundations and Applications
TL;DR: In this article, a mathematical background of Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractality products of fractal intersections of fractalities.
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Lévy processes and infinitely divisible distributions
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Book
The geometry of fractal sets
TL;DR: In this paper, a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension is presented, including questions of local density and the existence of tangents of such sets, and the dimensional properties of their projections in various directions.
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Some Random Series of Functions
TL;DR: A few tools from probability theory can be found in this paper, such as: 1. Random series in a Banach space, 2. Random Taylor series, 3. Random Fourier series, 4. Random point masses on the circle, 5. Gaussian variables and Gaussian series.