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Wavelet analysis of the multivariate fractional Brownian motion

Published online by Cambridge University Press:  06 August 2013

Jean-François Coeurjolly ,
Pierre-Olivier Amblard  and
Sophie Achard
Jean-François Coeurjolly
Affiliation:
Laboratory Jean Kuntzmann, Grenoble University, France GIPSAlab/CNRS, Grenoble University, France
Pierre-Olivier Amblard
Affiliation:
GIPSAlab/CNRS, Grenoble University, France Dept. of Mathematics&Statistics, University of Melbourne, Australia. Jean-Francois.Coeurjolly@upmf-grenoble.fr
Sophie Achard
Affiliation:
GIPSAlab/CNRS, Grenoble University, France
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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 behaviour 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|α. The behaviour 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.

Keywords

Multivariate fractional Brownian motionwavelet analysiscross-correlationcross-spectrum
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 592 - 604
Copyright
© EDP Sciences, SMAI, 2013

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