Asymptotic normality of wavelet covariances and of multivariate wavelet Whittle estimators

TL;DR

This paper proves the asymptotic normality of wavelet covariances and Whittle estimators for multivariate long-range dependent processes, with applications demonstrated on rat brain data.

Contribution

It establishes the asymptotic Gaussian behavior of wavelet covariances and the normality of wavelet-based Whittle estimators for non-Gaussian multivariate long-range processes.

Findings

Wavelet covariances are asymptotically Gaussian.

Wavelet Whittle estimators are asymptotically normal.

Explicit asymptotic covariance expressions are provided.

Abstract

Multivariate processes with long-range dependence properties can be encountered in many fields of application. Two fundamental characteristics in such frameworks are long-range dependence parameters and correlations between component time series. We consider multivariate long-range dependent linear processes, not necessarily Gaussian. We show that the covariances between the wavelet coefficients in this setting are asymptotically Gaussian. We also study the asymptotic distributions of the estimators of the long-range dependence parameter and the long-run covariance by a wavelet-based Whittle procedure. We prove the asymptotic normality of the estimators, and we provide an explicit expression for the asymptotic covariances. An empirical illustration of this result is proposed on a real dataset of rat brain connectivity.