On the validity of resampling methods under long memory

TL;DR

This paper investigates the validity of resampling methods for long-memory time series, providing bounds and conditions that ensure their asymptotic consistency despite strong dependence.

Contribution

It derives an efficient bound for the canonical correlation in long-memory series and establishes the asymptotic validity of subsampling methods under general conditions.

Findings

Bound for canonical correlation derived

Subsampling procedures shown to be consistent for long memory

Applications demonstrated for covariance, M-estimation, and empirical processes

Abstract

For long-memory time series, inference based on resampling is of crucial importance, since the asymptotic distribution can often be non-Gaussian and is difficult to determine statistically. However due to the strong dependence, establishing the asymptotic validity of resampling methods is nontrivial. In this paper, we derive an efficient bound for the canonical correlation between two finite blocks of a long-memory time series. We show how this bound can be applied to establish the asymptotic consistency of subsampling procedures for general statistics under long memory. It allows the subsample size $b$ to be $o(n)$, where $n$ is the sample size, irrespective of the strength of the memory. We are then able to improve many results found in the literature. We also consider applications of subsampling procedures under long memory to the sample covariance, M-estimation and empirical processes.