Normal approximation via non-linear exchangeable pairs

TL;DR

This paper introduces a novel functional analytic approach to Stein's method that improves normal and Gamma approximation bounds for various functionals, including U-statistics and subgraph counts, without requiring linear regression conditions.

Contribution

It develops a new exchangeable pairs technique that does not rely on approximate linear regression, leading to better bounds in normal approximation problems.

Findings

Better bounds for symmetric U-statistics.

Effective Wasserstein bounds for subgraph counts.

Improved normal approximation for Pearson's statistic.

Abstract

We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric $U$-statistics, we demonstrate in which respect this approach yields fundamentally better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic.