Central Limit Theorems for arrays of decimated linear processes

TL;DR

This paper establishes central limit theorems for decimated linear processes, enabling analysis of spectral density estimators and long-memory parameters in time-series, with implications for wavelet-based methods.

Contribution

It introduces CLTs for arrays of decimated linear processes, extending theoretical understanding and practical estimation techniques in time-series analysis.

Findings

CLTs for decimated linear processes established

Asymptotic behavior of spectral density estimators derived

Applications to long-memory parameter estimation using wavelets

Abstract

Linear processes are defined as a discrete-time convolution between a kernel and an infinite sequence of i.i.d. random variables. We modify this convolution by introducing decimation, that is, by stretching time accordingly. We then establish central limit theorems for arrays of squares of such decimated processes. These theorems are used to obtain the asymptotic behavior of estimators of the spectral density at specific frequencies. Another application, treated elsewhere, concerns the estimation of the long-memory parameter in time-series, using wavelets.