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 SAMOS - Statistique Appliquée et MOdélisation Stochastique

In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q>1 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 limit 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. 2007. ⟨hal-00184057v1⟩