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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/ps/2012011/
[1] , , , and , A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs. J. Neurosci. 26 (2006) 63-72.
[2] , , and , Fractal connectivity of long-memory networks. Phys. Rev. E 77 (2008) 036104.
[3] and , Identification of the multivariate fractional Brownian motion. IEEE Trans. Signal Process. 59 (2011) 5152-5168. | MR
[4] , , and , Basic properties of the multivariate fractional Brownian motion, edited by L. Chaumont. Séminaires et Congrès, Self-similar processes and their applications 28 (2012) 65-87. | MR | Zbl
[5] and , Cross-correlation of long range correlated series. J. Stat. Mech. (2009) P033037.
[6] , and , Drap brownien fractionnaire. Potential Anal. 17 (2002) 31-43. | MR | Zbl
[7] , , and , Wavelet estimator of long-range dependent processes. Stat. Inference Stoch. Process. 3 (2000) 85-99. | MR | Zbl
[8] and , Simulation of stationary Gaussian vector fields. Stat. Comput. 9 (1999) 265-268.
[9] , Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199-227. | MR | Zbl
[10] and , Integral representations of operator fractional Brownian motions. Bernouilli 17 (2011) 1-33. | MR | Zbl
[11] , , and , Estimators of long-memory: Fourier versus wavelets. J. Econ. 151 (2009) 159-177. | MR
[12] , On the spectrum of fractional Brownian motions. IEEE Trans. Inf. Theory 35 (1988) 197-199. | MR
[13] , Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans. Inf. Theory 38 (1992) 910-917. | MR | Zbl
[14] and , Generalized functions. Properties and Operations 1 (1964). | Zbl
[15] , A fractional multivariate long memory model for the US and the Canadian real output. Econ. Lett. 81 (2003) 355-359. | Zbl
[16] and , On the spectral density of the wavelet transform of fractional Brownian motions. J. Time Ser. Anal. 20 (1999) 560-563. | MR | Zbl
[17] , and , Covariance function of vector self-similar processes. Stat. Probab. Lett. 79 (2009) 2415-2421. | MR | Zbl
[18] and , Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422-437. | MR | Zbl
[19] and , Multifractional Brownian motion: definition and preliminary results. Rapport Recherche INRIA (1995).
[20] and , Correlation structure of the discrete wavelet coefficients of fractional Brownian motion. IEEE Trans. Inf. Theory 38 (1992) 904-909. | MR | Zbl
[21] and , Wavelet-based joint estimate of the long-range dependence parameters. IEEE Trans. Inf. Theory 45 (1999) 878-897. | MR | Zbl
[22] and , Inequalities for the rth absolute moment of a sum of random variables, 1 ≤ r ≤ 2. Ann. Math. Stat. 36 (1965) 299-303. | MR | Zbl
[23] , A Karhunen-Loève-like expansion for 1 / f processes via wavelets. IEEE Trans. Inf. Theory 36 (1990) 861-863.
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