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The $$n$$n-term Approximation of Periodic Generalized Lévy Processes

Author

Listed:
  • Julien Fageot

    (École polytechnique fédérale de Lausanne (EPFL))

  • Michael Unser

    (École polytechnique fédérale de Lausanne (EPFL))

  • John Paul Ward

    (North Carolina A&T State University)

Abstract

In this paper, we study the compressibility of random processes and fields, called generalized Lévy processes, that are solutions of stochastic differential equations driven by d-dimensional periodic Lévy white noises. Our results are based on the estimation of the Besov regularity of Lévy white noises and generalized Lévy processes. We show in particular that non-Gaussian generalized Lévy processes are more compressible in a wavelet basis than the corresponding Gaussian processes, in the sense that their $$n$$n-term approximation errors decay faster. We quantify this compressibility in terms of the Blumenthal–Getoor indices of the underlying Lévy white noise.

Suggested Citation

  • Julien Fageot & Michael Unser & John Paul Ward, 2020. "The $$n$$n-term Approximation of Periodic Generalized Lévy Processes," Journal of Theoretical Probability, Springer, vol. 33(1), pages 180-200, March.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:1:d:10.1007_s10959-018-00877-7
    DOI: 10.1007/s10959-018-00877-7
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    References listed on IDEAS

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    1. Davis, Richard A. & Knight, Keith & Liu, Jian, 1992. "M-estimation for autoregressions with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 145-180, February.
    2. Fageot, Julien & Fallah, Alireza & Unser, Michael, 2017. "Multidimensional Lévy white noise in weighted Besov spaces," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1599-1621.
    3. Ole E. Barndorff-Nielsen, 1997. "Processes of normal inverse Gaussian type," Finance and Stochastics, Springer, vol. 2(1), pages 41-68.
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