Kac's Central Limit Theorem by Stein's Method

TL;DR

This paper provides a new proof of Kac's Central Limit Theorem for dependent variables using Stein's method, demonstrating convergence to the normal distribution in the Wasserstein metric.

Contribution

It introduces a novel Stein's method-based proof for Kac's CLT, strengthening the convergence result to Wasserstein metric.

Findings

Convergence of normalized sample averages to normal distribution

Application of Stein's method to dependent variables

Stronger Wasserstein metric convergence

Abstract

In $1946$, Mark Kac proved a Central Limit type theorem for a sequence of random variables that were not independent. The random variables under consideration were obtained from the angle-doubling map. The idea behind Kac's proof was to show that although the random variables under consideration were not independent, they were what he calls \textit{statistically independent} (in modern terminology, this concept is called long range independence). The final conclusion of his paper was that the sample averages of the random variables, suitably normalized converges to the standard normal distribution. We describe a new proof of Mark Kac's result by applying Stein's method and show that the normalized sample averages converge to the standard normal distribution in the Wasserstein metric, which is stronger than the convergence in distribution.