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The \(n\)-term Approximation of Periodic Generalized Lévy Processes

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 \(n\)-term approximation errors decay faster. We quantify this compressibility in terms of the Blumenthal–Getoor indices of the underlying Lévy white noise.

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Notes

  1. 1.

    The cited works deal with general Lévy-type processes that do not necessarily have stationary increments.

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Acknowledgements

Funding was provided by the European Research Council (Grant No. 692726 - GlobalBioIm).

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Correspondence to John Paul Ward.

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This work is an extension of the conference paper [49].

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Fageot, J., Unser, M. & Ward, J.P. The \(n\)-term Approximation of Periodic Generalized Lévy Processes. J Theor Probab 33, 180–200 (2020). https://doi.org/10.1007/s10959-018-00877-7

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Keywords

  • Generalized Lévy processes
  • Lévy white noises
  • Besov regularity
  • n-term approximation
  • Compressibility

Mathematics Subject Classification (2010)

  • 60G20
  • 41A25