Wavelet estimation of the long memory parameter for Hermite polynomial of Gaussian processes^∗,∗∗

¹ Laboratoire Jean Kuntzmann, Université de Grenoble, CNRS, 38041 Grenoble Cedex 9. France
marianne.clausel@imag.fr
² Institut Telecom, Telecom Paris, CNRS LTCI, 46 rue Barrault, 75634 Paris Cedex 13, France
roueff@telecom-paristech.fr
³ Departement of Mathematics and Statistics, Boston University, Boston, MA 02215, USA
murad@math.bu.edu
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
Ciprian.Tudor@math.univ-lille1.fr

Received: 1 July 2011
Revised: 21 May 2012

Abstract

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.