A Berry-Esseen theorem for sample quantiles under association

TL;DR

This paper establishes a Berry-Esseen type theorem for sample quantiles of associated random variables, providing rates of normal approximation depending on covariance decay rates.

Contribution

It extends the Berry-Esseen theorem to associated variables, detailing how covariance decay influences the rate of convergence for sample quantiles.

Findings

Normal approximation rate is $O(n^{-1/2}\log^2 n)$ with exponential covariance decay.

Rate improves to $O(n^{-1/3})$ under polynomial covariance decay.

Provides theoretical bounds for the distribution of sample quantiles in dependent data.

Abstract

In this paper, the uniformly asymptotic normality for sample quantiles of associated random variables is investigated under some conditions on the decay of the covariances. We obtain the rate of normal approximation of order $O(n^{-1/2}\log^2 n)$ if the covariances decrease exponentially to $0$. The best rate is shown as $O(n^{-1/3})$ under a polynomial decay of the covariances.