Title :

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

Author(s) :

Clausel, Marianne [Auteur]
Statistique Apprentissage Machine [SAM]
Roueff, François [Auteur]
Laboratoire Traitement et Communication de l'Information [LTCI]
Taqqu, Murad [Auteur]
Department of Mathematics and Statistics [Boston]
Tudor, Ciprian [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

ALEA : Latin American Journal of Probability and Mathematical Statistics

Pages :

979­-1011

Publisher :

Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....]

Publication date :

2013-12-31

ISSN :

1980-0436

English keyword(s) :

Long--range dependence
Hermite processes
Wavelet coefficients
Wiener chaos
self-similar processes
Long--range dependence.

HAL domain(s) :

Mathématiques [math]/Probabilités [math.PR]
Mathématiques [math]/Statistiques [math.ST]
Statistiques [stat]/Théorie [stat.TH]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

ANR Project :

Systemes et Algorithmes Pervasifs au confluent des mondes physique et numérique

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL