Variations and estimators for the selfsimilarity order through Malliavin calculus

Ciprian A. Tudor (1, 2) , Frederi Viens (3)

1 CES - Centre d'économie de la Sorbonne
2 SAMOS - Statistique Appliquée et MOdélisation Stochastique
3 Department of Statistics [West Lafayette]

Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process. We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter $H$. Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all $H>1/2$, we show the remarkable fact that the process's data at time $1$ can be used to construct a distinct, compensated estimator with Gaussian asymptotics for $H\in(1/2,2/3)$.

Mots clés: multiple stochastic integrals; Malliavin calculus; Hermite process; fractional Brownian motion; non-central limit theorems; quadratic variations

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

Ciprian A. Tudor, Frederi Viens. Variations and estimators for the selfsimilarity order through Malliavin calculus. Annals of Probability, 2009, 37 (6), pp.2093-2134. ⟨10.1214/09-AOP459⟩. ⟨hal-00174730v3⟩