Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T06:47:15.221Z Has data issue: false hasContentIssue false
  • English
  • Français

Wavelet analysis of the multivariate fractionalBrownian 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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achard, S., Salvador, R., Whitcher, B., Suckling, J. and Bullmore, E., A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs. J. Neurosci. 26 (2006) 63–72. Google Scholar
Achard, S., Bassett, D.S., Meyer-Lindenberg, A. and Bullmore, E., Fractal connectivity of long-memory networks. Phys. Rev. E 77 (2008) 036104. Google Scholar
Amblard, P.O. and Coeurjolly, J.F., Identification of the multivariate fractional Brownian motion. IEEE Trans. Signal Process. 59 (2011) 5152–5168. Google Scholar
P.O. Amblard, J.F. Coeurjolly, F. Lavancier and A. Philippe, 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.
S. Arianos and A. Carbone, Cross-correlation of long range correlated series. J. Stat. Mech. (2009) P033037.
Ayache, A., Leger, S. and Pontier, M., Drap brownien fractionnaire. Potential Anal. 17 (2002) 31–43. Google Scholar
Bardet, J.M., Lang, G., Moulines, E. and Soulier, P., Wavelet estimator of long-range dependent processes. Stat. Inference Stoch. Process. 3 (2000) 85–99. Google Scholar
Chan, G. and Wood, A.T.A., Simulation of stationary Gaussian vector fields. Stat. Comput. 9 (1999) 265–268. Google Scholar
Coeurjolly, J.F., Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199–227. Google Scholar
Didier, G. and Pipiras, V., Integral representations of operator fractional Brownian motions. Bernouilli 17 (2011) 1–33. Google Scholar
Faÿ, G., Moulines, E., Roueff, F. and Taqqu, M.S., Estimators of long-memory: Fourier versus wavelets. J. Econ. 151 (2009) 159–177. Google Scholar
Flandrin, P., On the spectrum of fractional Brownian motions. IEEE Trans. Inf. Theory 35 (1988) 197–199. Google Scholar
Flandrin, P., Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans. Inf. Theory 38 (1992) 910–917. Google Scholar
I.M. Gel’fand and G.E. Shilov, Generalized functions. Properties and Operations 1 (1964).
Gil-Alana, L.A., A fractional multivariate long memory model for the US and the Canadian real output. Econ. Lett. 81 (2003) 355–359. Google Scholar
Kato, T. and Masry, E., On the spectral density of the wavelet transform of fractional Brownian motions. J. Time Ser. Anal. 20 (1999) 560–563. Google Scholar
Lavancier, F., Philippe, A. and Surgailis, D., Covariance function of vector self-similar processes. Stat. Probab. Lett. 79 (2009) 2415–2421. Google Scholar
Mandelbrot, B. and Van Ness, J., Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422–437. Google Scholar
R.F. Peltier and J. Lévy-Véhel, Multifractional Brownian motion: definition and preliminary results. Rapport Recherche INRIA (1995).
Tewfik, A.H. and Kim, M., Correlation structure of the discrete wavelet coefficients of fractional Brownian motion. IEEE Trans. Inf. Theory 38 (1992) 904–909. Google Scholar
Veitch, D. and Abry, P., Wavelet-based joint estimate of the long-range dependence parameters. IEEE Trans. Inf. Theory 45 (1999) 878–897. Google Scholar
von Bahr, B. and Esseen, C.G., Inequalities for the rth absolute moment of a sum of random variables, 1 ≤ r ≤ 2. Ann. Math. Stat. 36 (1965) 299–303. Google Scholar
Wornell, G.W., A Karhunen-Loève-like expansion for 1 / f processes via wavelets. IEEE Trans. Inf. Theory 36 (1990) 861–863. Google Scholar