## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 6 (2017), 2365-2399.

### On the validity of resampling methods under long memory

Shuyang Bai and Murad S. Taqqu

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 45, Number 6 (2017), 2365-2399.

**Dates**

Received: February 2016

Revised: August 2016

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1513328576

**Digital Object Identifier**

doi:10.1214/16-AOS1524

**Mathematical Reviews number (MathSciNet)**

MR3737895

**Zentralblatt MATH identifier**

06838136

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G09: Resampling methods

**Keywords**

Long memory long-range dependence resampling subsampling sampling window block sampling noncentral limit theorems canonical correlation

#### Citation

Bai, Shuyang; Taqqu, Murad S. On the validity of resampling methods under long memory. Ann. Statist. 45 (2017), no. 6, 2365--2399. doi:10.1214/16-AOS1524. https://projecteuclid.org/euclid.aos/1513328576