Stein's method for Conditional Central Limit Theorem

TL;DR

This paper extends Stein's method to analyze the rate of convergence in the Conditional Central Limit Theorem for jointly Gaussian variables, with applications to random sequences and graphs.

Contribution

It introduces a novel Stein's method approach for conditional convergence, extending classical CLT techniques to new conditional and multivariate settings.

Findings

Established a rate of convergence for conditional CLT using Stein's method.

Extended the approach to multivariate cases.

Applied the method to random binary sequences and Erdős-Rényi graphs.

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 Conditional Central Limit Theorem of the form $(X_n\mid Y_n=k)$, where $(X_n, Y_n)$ 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 Erd\"os-R\'enyi random graph.