U-statistics and random subgraph counts: Multivariate normal approximation via exchangeable pairs and embedding

TL;DR

This paper introduces an embedding method using exchangeable pairs for normal approximation, applied to U-statistics and subgraph counts in random graphs, overcoming previous linearity condition limitations.

Contribution

The authors extend the embedding approach to U-statistics and subgraph counts, enabling multivariate normal approximation without linearity constraints.

Findings

Successful application to U-statistics and subgraph counts

Improved normal approximation accuracy

Broader applicability of exchangeable pairs method

Abstract

In a recent paper by the authors, a new approach--called the "embedding method"--was introduced, which allows to make use of exchangeable pairs for normal and multivariate normal approximation with Stein's method in cases where the corresponding couplings do not satisfy a certain linearity condition. The key idea is to embed the problem into a higher dimensional space in such a way that the linearity condition is then satisfied. Here we apply the embedding to U-statistics as well as to subgraph counts in random graphs.