Electronic Journal of Probability

Wavelet analysis of the Besov regularity of Lévy white noise

Shayan Aziznejad and Julien Fageot

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Abstract

We characterize the local smoothness and the asymptotic growth rate of the Lévy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Lévy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-$\alpha $-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the Lévy white noise. Previous analyses only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires us to determine in which Besov spaces a given Lévy white noise is (almost surely) not. Our methods are based on the wavelet-domain characterization of Besov spaces and precise moment estimates for the wavelet coefficients of the Lévy white noise.

Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 158, 38 pp.

Dates
Received: 31 March 2020
Accepted: 11 November 2020
First available in Project Euclid: 24 December 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1608779099

Digital Object Identifier
doi:10.1214/20-EJP554

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 42C40: Wavelets and other special systems 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 60G20: Generalized stochastic processes

Keywords
Lévy white noise weighted Besov spaces wavelets moment estimates generalized random processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Aziznejad, Shayan; Fageot, Julien. Wavelet analysis of the Besov regularity of Lévy white noise. Electron. J. Probab. 25 (2020), paper no. 158, 38 pp. doi:10.1214/20-EJP554. https://projecteuclid.org/euclid.ejp/1608779099


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