A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter

Jean-Marc Bardet (1) , Ciprian A. Tudor (1, 2)

1 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
2 LPP - Laboratoire Paul Painlevé - UMR 8524

By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations.

Mots clés: Multiple Wiener-Itô integral; wavelet analysis; Rosenblatt process; fractional Brownian motion; noncentral limit theorem; self-similarity; parameter estimation

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Probabilités [math.PR]

Jean-Marc Bardet, Ciprian A. Tudor. A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter. Stochastic Analysis and Applications, 2010, 120, pp.2331-2362. ⟨10.1016/j.spa.2010.08.003⟩. ⟨hal-00339203v2⟩