Abstract
We characterize the local smoothness and the asymptotic growth rate of the Levy 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 Levy 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 Levy 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 Levy 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 Levy white noise.
Highlights
Introduction and Main ResultsWe study the Besov regularity of Lévy white noises
We are especially interested in identifying the critical local smoothness and the critical asymptotic growth rate of those random processes for any integrability parameter p ∈
We demonstrate that wavelet methods are adapted to the analysis of the Lévy white noise
Summary
We study the Besov regularity of Lévy white noises. We are especially interested in identifying the critical local smoothness and the critical asymptotic growth rate of those random processes for any integrability parameter p ∈ Regularity properties are usually stated in terms of the inclusion of the process in some weighted Besov spaces (positive result). It is worth noting that our analysis requires the identification of a new index associated to a Lévy white noise, characterized by moment properties. By relying on this index, our negative results suggest that some of the previous stateof-the-art inclusions are not sharp. The combination of positive and negative results allows us to determine the critical Besov parameters of a Lévy white noise, both for the local smoothness and the asymptotic behavior. Two consequences are the characterization of the critical Sobolev and Hölder-Zigmund regularities of the Lévy white noise in Corollary 1.2
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