# Wavelet analysis of the multivariate fractional brownian motion

• Volume: 17, page 592-604
• ISSN: 1292-8100

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## Abstract

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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.

## How to cite

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Coeurjolly, Jean-François, Amblard, Pierre-Olivier, and Achard, Sophie. "Wavelet analysis of the multivariate fractional brownian motion." ESAIM: Probability and Statistics 17 (2013): 592-604. <http://eudml.org/doc/273630>.

@article{Coeurjolly2013,
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.},
author = {Coeurjolly, Jean-François, Amblard, Pierre-Olivier, Achard, Sophie},
journal = {ESAIM: Probability and Statistics},
keywords = {multivariate fractional brownian motion; wavelet analysis; cross-correlation; cross-spectrum; multivariate fractional Brownian motion},
language = {eng},
pages = {592-604},
publisher = {EDP-Sciences},
title = {Wavelet analysis of the multivariate fractional brownian motion},
url = {http://eudml.org/doc/273630},
volume = {17},
year = {2013},
}

TY - JOUR
AU - Coeurjolly, Jean-François
AU - Amblard, Pierre-Olivier
AU - Achard, Sophie
TI - Wavelet analysis of the multivariate fractional brownian motion
JO - ESAIM: Probability and Statistics
PY - 2013
PB - EDP-Sciences
VL - 17
SP - 592
EP - 604
AB - 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.
LA - eng
KW - multivariate fractional brownian motion; wavelet analysis; cross-correlation; cross-spectrum; multivariate fractional Brownian motion
UR - http://eudml.org/doc/273630
ER -

## References

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