A Berry--Esseen theorem for sample quantiles under weak dependence

TL;DR

This paper establishes a Berry--Esseen theorem for sample quantiles of strongly-mixing random variables, providing a rate of convergence that is significant for applications in finance and econometrics dealing with heavy-tailed data.

Contribution

It proves a new Berry--Esseen bound for sample quantiles under polynomial mixing, with a convergence rate of $O(n^{-1/2})$, contrasting previous results for sample means.

Findings

Rate of normal approximation is $O(n^{-1/2})$ for sample quantiles.

Results apply to heavy-tailed financial time series data.

Enhances understanding of quantile-based methods in econometrics.

Abstract

This paper proves a Berry--Esseen theorem for sample quantiles of strongly-mixing random variables under a polynomial mixing rate. The rate of normal approximation is shown to be $O(n^{-1/2})$ as $n\to\infty$, where $n$ denotes the sample size. This result is in sharp contrast to the case of the sample mean of strongly-mixing random variables where the rate $O(n^{-1/2})$ is not known even under an exponential strong mixing rate. The main result of the paper has applications in finance and econometrics as financial time series data often are heavy-tailed and quantile based methods play an important role in various problems in finance, including hedging and risk management.