Stein couplings for normal approximation

TL;DR

This paper introduces Stein couplings, a new framework within Stein's method, enabling routine normal approximation across various complex probabilistic models with diverse applications.

Contribution

It presents the concept of Stein couplings, unifying and extending existing approaches for normal approximation, with broad applicability demonstrated through multiple examples.

Findings

Normal approximation becomes routine with Stein couplings.

Versatile applications in combinatorics, occupancy schemes, and random graphs.

Introduction of new non-standard couplings for complex models.

Abstract

In this article we propose a general framework for normal approximation using Stein's method. We introduce the new concept of Stein couplings and we show that it lies at the heart of popular approaches such as the local approach, exchangeable pairs, size biasing and many other approaches. We prove several theorems with which normal approximation for the Wasserstein and Kolmogorov metrics becomes routine once a Stein coupling is found. To illustrate the versatility of our framework we give applications in Hoeffding's combinatorial central limit theorem, functionals in the classic occupancy scheme, neighbourhood statistics of point patterns with fixed number of points and functionals of the components of randomly chosen vertices of sub-critical Erdos-Renyi random graphs. In all these cases, we use new, non-standard couplings.