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

Nourdin, Ivan; Nualart, David; Tudor, Ciprian A.

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 1055-1079.

Dans ce papier, nous prouvons des théorèmes de la limite centrale et non-centrale pour les variations à poids d'ordre q du mouvement brownien fractionnaire d'indice H∈(0, 1), pour q un entier supérieur ou égal à 2. Il y a trois cas, suivant la position de H par rapport à 1/2q et 1-1/2q. Si 1/2q<H≤1-1/2q, nous montrons un théorème de la limite centrale vers une variable aléatoire de loi conditionnellement gaussienne. Si H<1/2q, nous montrons la convergence dans L2 vers une limite qui dépend seulement du mouvement brownien fractionnaire. Si H>1-1/2q, nous montrons la convergence dans L2 vers une intégrale stochastique par rapport au processus d'Hermite d'ordre q.

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∈(0, 1), where q is an integer. The central limit holds for 1/2q<H≤1-1/2q, the limit being a conditionally gaussian distribution. If H<1/2q we show the convergence in L2 to a limit which only depends on the fractional brownian motion, and if H>1-1/2q we show the convergence in L2 to a stochastic integral with respect to the Hermite process of order q.

Classification : 60F05, 60H05, 60G15, 60H07
Mots clés : fractional brownian motion, central limit theorem, non-central limit theorem, Hermite process

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