HAL: hal-00501720, version 1

arXiv: 1007.2109

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v1 (2010-07-13)

v2 (2011-06-30)

Normalized causal and well-balanced multivariate fractional Brownian motion

Jean-François Coeurjolly 1, 2, Pierre-Olivier Amblard 1, Sophie Achard 1

(2010-07-10)

This paper is devoted to study some properties of an extension of the well-known fractional Brownian motion to the multivariate case. Following recent works from Lavancier et. al., we study the covariance structure of the multivariate fractional Gaussian noise. We evaluate several parameters of the model that allow to control the correlation structure at lag zero between all the components of the multivariate process. We particularly focus on two cases for which we can relate characteristic parameters of the covariance function to parameters of the stochastic representation of the processes. These cases are the causal case, a direct multivariate generalization of Mandelbrot\&Van Ness representation, and the well-balanced case which adds to the previous case an anti-causal filtering of a Brownian motion. The characterization of the covariance function is then used to study the multivariate fractional Gaussian noise, defined as the increment process of the multivariate fractional Brownian motion. We study the covariance structure as well as the spectral structure of this multivariate stationary process. We exhibit the intriguing facts that two fractional Gaussian noise may be long-range interdependent when only one is long-range dependent. We then perform a wavelet analysis of the multivariate fractional Brownian motion, and show that the wavelet analysis may destroy the long-range interdependence if the wavelet is properly chosen.

1:	Grenoble Images Parole Signal Automatique (GIPSA-lab)
	CNRS : UMR5216 – Université Joseph Fourier - Grenoble I – Université Pierre-Mendès-France - Grenoble II – Université Stendhal - Grenoble III – Institut Polytechnique de Grenoble - Grenoble Institute of Technology
2:	Laboratoire Jean Kuntzmann (LJK)
	CNRS : UMR5224 – Université Joseph Fourier - Grenoble I – Université Pierre-Mendès-France - Grenoble II – Institut Polytechnique de Grenoble - Grenoble Institute of Technology

Subject

:

Mathematics/Statistics Statistics/Statistics Theory Engineering Sciences/Signal and Image processing Computer Science/Signal and Image Processing

Keyword(s): multivariate fractional Brownian motion – cross-covariance – cross-spectral density – wavelet analysis

hal-00501720, version 1

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From: Jean-François Coeurjolly <>

Submitted on: Monday, 12 July 2010 15:35:49

Updated on: Thursday, 29 July 2010 15:52:44