Wavelet Analysis of the Besov Regularity of L\'evy White Noises

Abstract

We characterize the local smoothness and the asymptotic growth rate of the L\'evy 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 L\'evy 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 L\'evy 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 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 L\'evy white noise.