Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Ivan Nourdin, David Nualart, and Ciprian A. Tudor

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 4 (2010), 1055-1079.

Dates
First available in Project Euclid: 4 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1288878338

Digital Object Identifier
doi:10.1214/09-AIHP342

Mathematical Reviews number (MathSciNet)
MR2744886

Zentralblatt MATH identifier
1221.60031

Subjects
Primary: 60F05: Central limit and other weak theorems 60H05: Stochastic integrals 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Fractional Brownian motion Central limit theorem Non-central limit theorem Hermite process

Citation

Nourdin, Ivan; Nualart, David; Tudor, Ciprian A. Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 1055--1079. doi:10.1214/09-AIHP342. https://projecteuclid.org/euclid.aihp/1288878338


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