Quantitative CLTs for symmetric $U$-statistics using contractions

TL;DR

This paper establishes quantitative central limit theorems for symmetric U-statistics using contraction operators, extending classical results and providing explicit bounds relevant for geometric and graph-based applications.

Contribution

It introduces a new approach to CLTs for symmetric U-statistics via contraction operators, applicable in multi-dimensional and non-Hoeffding-degenerate cases, with explicit bounds for geometric applications.

Findings

Derived explicit CLT bounds for subgraph counts in random geometric graphs.

Extended classical CLTs to the dense parameter regime for uniformly distributed points.

Provided new qualitative CLTs for subgraph counting in Euclidean spaces.

Abstract

We consider sequences of symmetric $U$-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of {\it contraction operators}. Our results represent an explicit counterpart to analogous criteria that are available for sequences of random variables living on the Gaussian, Poisson or Rademacher chaoses, and are perfectly tailored for geometric applications. As a demonstration of this fact, we develop explicit bounds for subgraph counting in generalised random graphs on Euclidean spaces; special attention is devoted to the so-called `dense parameter regime' for uniformly distributed points, for which we deduce CLTs that are new even in their qualitative statement, and that substantially extend classical findings by Jammalamadaka and Janson (1986) and Bhattacharaya and Ghosh (1992).