Central and non-central limit theorems for weighted power variations of fractional Brownian motion

Ivan Nourdin (1) , David Nualart (2) , Ciprian A. Tudor (3, 4)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Department of Mathematics [Lawrence, Kansas]
3 CES - Centre d'économie de la Sorbonne
4 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)

In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q>=2 of the fractional Brownian motion with Hurst parameter H in (0,1), where q is an integer. The central limit holds for 1/(2q) 1-1/(2q), we show the convergence in L^2 to a stochastic integral with respect to the Hermite process of order q.

Mots clés: fractional Brownian motion; central limit theorem; non-central limit theorem; Hermite process

Domaines: Probabilités [math.PR]

Ivan Nourdin, David Nualart, Ciprian A. Tudor. Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2010, 46 (4), pp.1055-1079. ⟨10.1214/09-AIHP342⟩. ⟨hal-00184057v3⟩