On normal approximations for the two-sample problem on multidimensional tori

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. (2011) and further developed by Peccati and Zheng (2010) and Bourguin and Peccati (2014). Particular cases of the proposed framework allow to consider the two-sample problem on the circle as well as the local two-sample problem on Rq through a local homeomorphism argument.