Wavelet analysis of the multivariate fractional Brownian motion

TL;DR

This paper investigates the wavelet analysis of multivariate fractional Brownian motion, revealing how wavelet properties influence correlation and cross-spectral density, especially regarding long-range dependence and zero-frequency behavior.

Contribution

It provides a detailed analysis of the correlation structure and cross-spectral density of mfBm through wavelet transforms, including conditions for eliminating long-range dependence.

Findings

Wavelet transform can eliminate long-range dependence with sufficient null moments.

The cross-spectral density exists and is explicitly evaluated.

Asymptotic analysis confirms the behavior at zero frequency.

Abstract

The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behavior of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral density is also considered in a second part. Its existence is proved and its evaluation is performed using a von Bahr-Essen like representation of the function $\sign(t) |t|^\alpha$. The behavior of the cross-spectral density of the wavelet field at the zero frequency is also developed and confirms the results provided by the asymptotic analysis of the correlation.