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

Authors :
Clausel, Marianne
Roueff, François
Taqqu, Murad S.
Tudor, Ciprian A.
Publication Year :
2012

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-It\^o 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-It\^o integral of order greater than two.

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.1201.4831
Document Type :
Working Paper