Normal Limits, Nonnormal Limits, and the Bootstrap for Quantiles of Dependent Data

TL;DR

This paper investigates the asymptotic behavior of quantiles for dependent data, establishing conditions for the central limit theorem and bootstrap consistency, including cases with non-normal limits.

Contribution

It provides weak and strong consistency results for the bootstrap of quantiles under dependence, extending existing theory to less restrictive conditions.

Findings

CLT for quantiles under weak dependence

Weak bootstrap consistency under mild conditions

Potential non-normal limits without differentiability

Abstract

We will show under very weak conditions on differentiability and dependence that the central limit theorem for quantiles holds and that the block bootstrap is weakly consistent. Under slightly stronger conditions, the bootstrap is strongly consistent. Without the differentiability condition, quantiles might have a non-normal asymptotic distribution and the bootstrap might fail.