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

1 SAM - Statistique Apprentissage Machine
LJK - Laboratoire Jean Kuntzmann
Abstract : 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.
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https://hal-imt.archives-ouvertes.fr/hal-00662317
Contributor : François Roueff <>
Submitted on : Wednesday, November 27, 2013 - 10:02:15 AM
Last modification on : Friday, July 19, 2019 - 2:57:12 PM
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• HAL Id : hal-00662317, version 2
• ARXIV : 1201.4831

### Citation

Marianne Clausel, François Roueff, Murad Taqqu, Ciprian Tudor. High order chaotic limits of wavelet scalograms under long--range dependence. ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada, 2013, 10 (2), pp.979­-1011. ⟨hal-00662317v2⟩

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