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

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.

Published in at http://dx.doi.org/10.1214/08-AAP533 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)