Stein couplings for normal approximation

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.

54 pages; minor changes and corrections

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• Normal approximation under local dependence
• A new method of normal approximation
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• Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing
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• Discretized normal approximation by Stein's method
• Rates of convergence for multivariate normal approximation with applications to dense graphs and doubly indexed permutation statistics
• A Central Limit Theorem for Vincular Permutation Patterns
• Cramér-type Moderate Deviation Theorems for Nonnormal Approximation
• Central limit theorems in the configuration model
• A Berry-Esseen bound for the uniform multinomial occupancy model
• Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field
• Palm theory, random measures and Stein couplings
• Central moment inequalities using Stein's method
• Local limit theorems for occupancy models
• Skewness correction in tail probability approximations for sums of local statistics
• Error bounds in local limit theorems using Stein's method
• A multivariate CLT for bounded decomposable random vectors with the best known rate
• Stein meets Malliavin in normal approximation
• Fundamentals of Stein's method
• A geometric approach to conjugation-invariant random permutations