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.
Mots-clés : multivariate fractional brownian motion, wavelet analysis, cross-correlation, cross-spectrum
@article{PS_2013__17__592_0,
author = {Coeurjolly, Jean-Fran\c{c}ois and Amblard, Pierre-Olivier and Achard, Sophie},
title = {Wavelet analysis of the multivariate fractional brownian motion},
journal = {ESAIM: Probability and Statistics},
pages = {592--604},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2012011},
mrnumber = {3085634},
zbl = {1293.42038},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2012011/}
}
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 SP - 592 EP - 604 VL - 17 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2012011/ DO - 10.1051/ps/2012011 LA - en ID - PS_2013__17__592_0 ER -
%0 Journal Article %A Coeurjolly, Jean-François %A Amblard, Pierre-Olivier %A Achard, Sophie %T Wavelet analysis of the multivariate fractional brownian motion %J ESAIM: Probability and Statistics %D 2013 %P 592-604 %V 17 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2012011/ %R 10.1051/ps/2012011 %G en %F PS_2013__17__592_0
Coeurjolly, Jean-François; Amblard, Pierre-Olivier; Achard, Sophie. Wavelet analysis of the multivariate fractional brownian motion. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 592-604. doi : 10.1051/ps/2012011. http://www.numdam.org/articles/10.1051/ps/2012011/
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