Ricerc@Sapienza

In this paper, quantitative central limit theorems for U-statistics on the q-dimensional torus defined in the framework of the two-sample problem for Poisson processes are derived. In particular, the U-statistics are built over tight frames defined by wavelets, named toroidal needlets, enjoying excellent localization properties in both harmonic and frequency domains. The rates of convergence to Gaussianity for these statistics are obtained by means of the so-called Stein–Malliavin techniques on the Poisson space, as introduced by Peccati et al.

Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients exactly, in theory and practice (to machine precision). Scale-discretised wavelets are closely related to spherical needlets (both were developed independently at about the same time) but relax the axisymmetric property of needlets so that directional signal content can be probed.

In this paper we study the tail behavior of Mexican needlets, a class of spherical wavelets introduced by Geller and Mayeli in [12]. More specifically, we provide an explicit upper bound depending on the resolution level j and a parameter s governing the shape of the Mexican needlets.