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

Marianne Clausel (1) , François Roueff (2) , Murad S. Taqqu (3) , Ciprian A. Tudor (4)

1 SAM - Statistique Apprentissage Machine
2 LTCI - Laboratoire Traitement et Communication de l'Information
3 BU - Boston University [Boston]
4 LPP - Laboratoire Paul Painlevé

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.

Keywords: Hermite processes; Wavelet coefficients; Wiener chaos; self-similar processes; Long--range dependence

Domains: Mathematics [math]; Statistics [math.ST]; Statistics [stat]; Statistics Theory [stat.TH]

Marianne Clausel, François Roueff, Murad S. Taqqu, Ciprian A. Tudor. Wavelet estimation of the long memory parameter for Hermite polynomial of Gaussian processes. ESAIM: Probability and Statistics, 2014, 18, pp.42-76. ⟨10.1051/ps/2012026⟩. ⟨hal-00590798v3⟩