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

23 January 2012
Marianne Clausel
François Roueff
M. Taqqu
C. Tudor
ArXiv (abs)PDFHTML
Abstract

Let GGG be a non--linear function of a Gaussian process {Xt}t∈Z\{X_t\}_{t\in\mathbb{Z}}{Xt​}t∈Z​ with long--range dependence. The resulting process {G(Xt)}t∈Z\{G(X_t)\}_{t\in\mathbb{Z}}{G(Xt​)}t∈Z​ is not Gaussian when GGG is not linear. We consider random wavelet coefficients associated with {G(Xt)}t∈Z\{G(X_t)\}_{t\in\mathbb{Z}}{G(Xt​)}t∈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 GGG 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 GGG 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.

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