HAL: hal-00662317, version 1

arXiv: 1201.4831

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High order chaotic limits of wavelet scalograms under long--range dependence

Marianne Clausel 1, François Roueff 2, Murad S. Taqqu 3, Ciprian A. Tudor 4

(2012-01-23)

Let $G$ be a non--linear function of a Gaussian process $\{X_t\}_{t\in\mathbb{Z}}$ with long--range dependence. The resulting process $\{G(X_t)\}_{t\in\mathbb{Z}}$ is not Gaussian when $G$ is not linear. We consider random wavelet coefficients associated with $\{G(X_t)\}_{t\in\mathbb{Z}}$ and the corresponding wavelet scalogram which is the average of squares of wavelet coefficients over locations. We obtain the asymptotic behavior of the scalogram as the number of observations and scales tend to infinity. It is known that when $G$ is a Hermite polynomial of any order, then the limit is either the Gaussian or the Rosenblatt distribution, that is, the limit can be represented by a multiple Wiener-Itô integral of order one or two. We show, however, that there are large classes of functions $G$ which yield a higher order Hermite distribution, that is, the limit can be represented by a a multiple Wiener-Itô integral of order greater than two.

1:	Laboratoire Jean Kuntzmann (LJK)
	CNRS : UMR5224 – Université Joseph Fourier - Grenoble I – Université Pierre-Mendès-France - Grenoble II – Institut Polytechnique de Grenoble - Grenoble Institute of Technology
2:	Laboratoire Traitement et Communication de l'Information [Paris] (LTCI)
	Télécom ParisTech – CNRS : UMR5141
3:	Department of Mathematics and Statistics [Boston]
	Boston University
4:	Laboratoire Paul Painlevé (LPP)
	CNRS : UMR8524 – Université Lille I - Sciences et technologies

Subject

:

Mathematics/Probability Mathematics/Statistics Statistics/Statistics Theory

Keyword(s): Hermite processes – Wavelet coefficients – Wiener chaos – self-similar processes – Long--range dependence.

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From: François Roueff <>

Submitted on: Monday, 23 January 2012 17:12:49

Updated on: Monday, 23 January 2012 20:37:57