Block bootstrap for the empirical process of long-range dependent data

TL;DR

This paper investigates the behavior of the bootstrap method applied to long-range dependent data, revealing that it converges to a Gaussian limit, which may cause failures in certain noncentral limit scenarios.

Contribution

It demonstrates that the bootstrap empirical process for long-range dependent data converges to a Gaussian limit, highlighting potential limitations in noncentral limit cases.

Findings

Bootstrap empirical process converges to a Gaussian limit.

Failure of bootstrap in noncentral limit theorem cases.

Limit is semi-degenerate with a Gaussian random part.

Abstract

We consider long-range dependent data. It is shown that the bootstrapped empirical process of these data converges to a semi-degenerate limit. The random part of this limit is always Gaussian. Thus the bootstrap might fail when the original empirical process accomplishes a noncentral limit theorem.