Volume 18, 2014
|Page(s)||42 - 76|
|Published online||28 November 2013|
Wavelet estimation of the long memory parameter for Hermite polynomial of Gaussian processes∗,∗∗
Laboratoire Jean Kuntzmann, Université de Grenoble,
Grenoble Cedex 9.
2 Institut Telecom, Telecom Paris, CNRS LTCI, 46 rue Barrault, 75634 Paris Cedex 13, France
3 Departement of Mathematics and Statistics, Boston University, Boston, MA 02215, USA
4 Laboratoire Paul Painlevé, UMR 8524 du CNRS, Université Lille 1, 59655 Villeneuve d’Ascq, France. Associate member: SAMM, Université de Panthéon-Sorbonne Paris 1
Revised: 21 May 2012
We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long–memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener–Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.
Mathematics Subject Classification: 42C40 / 60G18 / 62M15 / 60G20 / 60G22
Key words: Hermite processes / wavelet coefficients / wiener chaos / self-similar processes / long–range dependence
© EDP Sciences, SMAI, 2013
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