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Wavelet estimation of the long memory parameter for Hermitepolynomial of Gaussian processes∗∗∗

Published online by Cambridge University Press:  28 November 2013

M. Clausel ,
F. Roueff ,
M.S. Taqqu  and
C. Tudor
M. Clausel
Affiliation:
Laboratoire Jean Kuntzmann, Université de Grenoble, CNRS, 38041 Grenoble Cedex 9. France. marianne.clausel@imag.fr
F. Roueff
Affiliation:
Institut Telecom, Telecom Paris, CNRS LTCI, 46 rue Barrault, 75634 Paris Cedex 13, France; roueff@telecom-paristech.fr
M.S. Taqqu
Affiliation:
Departement of Mathematics and Statistics, Boston University, Boston, MA 02215, USA; murad@math.bu.edu
C. Tudor
Affiliation:
Laboratoire Paul Painlevé, UMR 8524 du CNRS, Université Lille 1, 59655 Villeneuve d’Ascq, France. Associate member: SAMM, Université de Panthéon-Sorbonne Paris 1; Ciprian.Tudor@math.univ-lille1.fr
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Abstract

We consider stationary processes with long memory which are non-Gaussian and representedas Hermite polynomials of a Gaussian process. We focus on the corresponding waveletcoefficients and study the asymptotic behavior of the sum of their squares since this sumis often used for estimating the long–memory parameter. We show that the limit is notGaussian but can be expressed using the non-Gaussian Rosenblatt process defined as aWiener–Itô integral of order 2. This happens even if the original process is definedthrough a Hermite polynomial of order higher than 2.

Keywords

Hermite processeswavelet coefficientswiener chaosself-similar processeslong–range dependence
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 42 - 76
Copyright
© EDP Sciences, SMAI, 2013

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References

Abry, P. and Pipiras, V., Wavelet-based synthesis of the Rosenblatt process. Eurasip Signal Processing 86 (2006) 2326–2339. Google Scholar
Abry, P. and Veitch, D., Wavelet analysis of long–range-dependent traffic. IEEE Trans. Inform. Theory 44 (1998) 2–15. Google Scholar
Abry, P., Veitch, D. and Flandrin, P., Long-range dependence: revisiting aggregation with wavelets. J. Time Ser. Anal. 19 (1998) 253–266. ISSN 0143-9782. Google Scholar
Abry, P., Helgason, H. and V., Pipiras, Wavelet-based analysis of non-Gaussian long–range dependent processes and estimation of the Hurst parameter. Lithuanian Math. J. 51 (2011) 287–302. Google Scholar
Bardet, J.-M., Statistical study of the wavelet analysis of fractional Brownian motion. IEEE Trans. Inform. Theory 48 (2002) 991–999. Google Scholar
Bardet, J.-M. and Tudor, C.A., A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter. Stochastic Process. Appl. 120 (2010) 2331–2362. Google Scholar
Bardet, J.-M., Lang, G., Moulines, E. and Soulier, P., Wavelet estimator of long–range dependent processes. 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998). Stat. Inference Stoch. Process. 3 (2000) 85–99. Google Scholar
Bardet, J.M., Bibi, H. and Jouini, A., Adaptive wavelet based estimator of the memory parameter for stationary gaussian processes. Bernoulli 14 (2008) 691–724. Google Scholar
Breton, J.-C. and Nourdin, I., Error bounds on the non-normal approximation of hermite power variations of fractional brownian motion. Electron. Commun. Probab. 13 (2008) 482–493. Google Scholar
Chronopoulou, A., Tudor, C. and Viens, F., Self-similarity parameter estimation and reproduction property for non-gaussian Hermite processes. Commun. Stoch. Anal. 5 (2011) 161–185. Google Scholar
Clausel, M., Roueff, F., Taqqu, M.S. and Tudor, C., Large scale behavior of wavelet coefficients of non-linear subordinated processes with long memory. Appl. Comput. Harmonic Anal. 32 (2012) 223–241. Google Scholar
M. Clausel, F. Roueff, M.S. Taqqu and C. Tudor, High order chaotic limits of wavelet scalograms under long–range dependence. Technical report, Hal–Institut Telecom (2012). http://hal-institut-telecom.archives-ouvertes.fr/hal-00662317.
Dobrushin, R.L. and Major, P., Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 (1979) 27–52. Google Scholar
P. Embrechts and M. Maejima, Selfsimilar processes. Princeton University Press, Princeton, New York (2002).
Flandrin, P., On the spectrum of fractional Brownian motions. IEEE Trans. Inform. Theory IT-35 (1989) 197–199. Google Scholar
P. Flandrin, Some aspects of nonstationary signal processing with emphasis on time-frequency and time-scale methods. Edited by J.M. Combes, A. Grossman and Ph. Tchamitchian, Wavelets. Springer-Verlag (1989) 68–98.
P. Flandrin, Fractional Brownian motion and wavelets. Edited by M. Farge, J.C.R. Hung and J.C. Vassilicos, Fractals and Fourier Transforms-New Developments and New Applications. Oxford University Press (1991).
P. Flandrin, Time-Frequency/Time-scale Analysis, 1st edition. Academic Press (1999).
Fox, R. and Taqqu, M.S.. Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 (1986) 517–532. Google Scholar
Giraitis, L. and Surgailis, D., Central limit theorems and other limit theorems for functionals of gaussian processes. Z. Wahrsch. verw. Gebiete 70 (1985) 191–212. Google Scholar
Giraitis, L. and Taqqu, M.S., Whittle estimator for finite-variance non-gaussian time series with long memory. Ann. Statist. 27 (1999) 178–203. Google Scholar
Lawrance, A.J. and Kottegoda, N.T., Stochastic modelling of riverflow time series. J. Roy. Statist. Soc. Ser. A 140 (1977) 1–47. Google Scholar
P. Major, Multiple Wiener-Itô integrals, vol. 849 of Lect. Notes Math. Springer, Berlin (1981).
Moulines, E., Roueff, F. and Taqqu, M.S., On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter. J. Time Ser. Anal. 28 (2007) 155–187. Google Scholar
Nourdin, I. and Peccati, G., Stein’s method meets Malliavin calculus: a short survey with new estimates. Technical report, Recent Advances in Stochastic Dynamics and Stochastic Analysis 8 (2010) 207–236. Google Scholar
Nourdin, I. and Peccati, G., Stein’s method on wiener chaos. Probability Theory and Related Fields 154 (2009) 75–118. Google Scholar
D. Nualart, The Malliavin Calculus and Related Topics. Springer (2006).
Robinson, P.M., Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 (1995) 1048–1072. Google Scholar
Robinson, P.M., Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 (1995) 1630–1661. Google Scholar
Roueff, F. and Taqqu, M. S., Central limit theorems for arrays of decimated linear processes. Stoch. Proc. Appl. 119 (2009) 3006–3041. Google Scholar
Roueff, F. and Taqqu, M.S., Asymptotic normality of wavelet estimators of the memory parameter for linear processes. J. Time Ser. Anal. 30 (2009) 534–558.Google Scholar
A. Scherrer, Analyses statistiques des communications sur puce. Ph.D. thesis, École normale supérieure de Lyon (2006). Available on http://www.ens-lyon.fr/LIP/Pub/Rapports/PhD/PhD2006/PhD2006-09.pdf.
Taqqu, M.S., A representation for self-similar processes. Stoch. Proc. Appl. 7 (1978) 55–64. Google Scholar
Taqqu, M.S., Central limit theorems and other limit theorems for functionals of gaussian processes. Z. Wahrsch. verw. Gebiete 70 (1979) 191–212. Google Scholar
Wornell, G.W. and Oppenheim, A.V., Estimation of fractal signals from noisy measurements using wavelets. IEEE Trans. Signal Process. 40 (1992) 611–623. Google Scholar