Central and non-central limit theorems for weighted power variations of fractional brownian motion
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, p. 1055-1079

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.

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.

DOI : https://doi.org/10.1214/09-AIHP342
Classification:  60F05,  60H05,  60G15,  60H07
Keywords: fractional brownian motion, central limit theorem, non-central limit theorem, Hermite process
@article{AIHPB_2010__46_4_1055_0,
     author = {Nourdin, Ivan and Nualart, David and Tudor, Ciprian A.},
     title = {Central and non-central limit theorems for weighted power variations of fractional brownian motion},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {4},
     year = {2010},
     pages = {1055-1079},
     doi = {10.1214/09-AIHP342},
     zbl = {1221.60031},
     mrnumber = {2744886},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_4_1055_0}
}
Nourdin, Ivan; Nualart, David; Tudor, Ciprian A. Central and non-central limit theorems for weighted power variations of fractional brownian motion. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 4, pp. 1055-1079. doi : 10.1214/09-AIHP342. http://www.numdam.org/item/AIHPB_2010__46_4_1055_0/

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