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

TL;DR

This paper develops quantitative central limit theorems for U-statistics on multidimensional tori, using wavelet-based needlets and Stein-Malliavin techniques to analyze normal approximations in two-sample Poisson process problems.

Contribution

It introduces a novel framework for two-sample problems on tori using wavelet needlets and Stein-Malliavin methods for precise normal approximation bounds.

Findings

Derived Berry-Esseen bounds for U-statistics on tori.

Extended the framework to circle and local Euclidean two-sample problems.

Provided explicit normal approximation results with localization properties.

Abstract

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 Berry-Ess\'een type bounds associated with the normal approximations for these statistics are obtained by means of the so-called Stein-Malliavin techniques on the Poisson space, as introduced by Peccati, Sol\'e, Taqqu, Utzet (2011) and further developed by Peccati, Zheng (2010) and Bourguin, 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 $\mathbb{R}^q$ through a local homeomorphism argument.