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multivariate fractional brownian motion

ESAIM: Probability and Statistics (1)

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Wavelet analysis of the multivariate fractionalBrownian motion
Jean-François Coeurjolly, Pierre-Olivier Amblard, Sophie Achard
Journal:

ESAIM: Probability and Statistics / Volume 17 / 2013

Published online by Cambridge University Press:

06 August 2013, pp. 592-604

Print publication:

2013
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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.