Stein’s method for conditional central limit theorem

Abstract

In the seventies, Charles Stein revolutionized the way of proving the central limit theorem by introducing a method that utilizes a characterization equation for Gaussian distribution. In the last 50 years, much research has been done to adapt and strengthen this method to a variety of different settings and other limiting distributions. However, it has not been yet extended to study conditional convergences. In this article we develop a novel approach, using Stein’s method for exchangeable pairs, to find a rate of convergence in the conditional central limit theorem of the form $({\mathit{X}}_{\mathit{n}}\mid {\mathit{Y}}_{\mathit{n}}=\mathit{k})$, where $({\mathit{X}}_{\mathit{n}},{\mathit{Y}}_{\mathit{n}})$ are asymptotically jointly Gaussian, and extend this result to a multivariate version. We apply our general result to several concrete examples, including pattern count in a random binary sequence and subgraph count in Erdős–Rényi random graph.