Wavelet Analysis of the Besov Regularity of Lévy White Noises

Shayan Aziznejad, Julien Fageot, Michael Unser

Published 2018-01-28Version 1

In this paper, we characterize the local smoothness and the asymptotic growth rate of L\'evy white noises. We do so by identifying the weighted Besov spaces in which they are localized. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the L\'evy white noise. We deduce the critical local smoothness when the two indices coincide, which is true for symmetric-alpha-stable, compound Poisson and symmetric-gamma white noises. Second, we express the critical asymptotic growth rate in terms of the moments properties of the L\'evy white noise. Previous analysis only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires to determine in which Besov spaces a given L\'evy 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 noise.