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

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.

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

[1] P. Breuer and P. Major. Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 (1983) 425-441. | Zbl | MR

[2] K. Burdzy and J. Swanson. A change of variable formula with Itô correction term. Preprint, 2008. Available at arXiv:0802.3356. | Zbl | MR

[3] P. Cheridito and D. Nualart. Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H in (0, 1/2). Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 1049-1081. | Zbl | MR | Numdam

[4] J. M. Corcuera, D. Nualart and J. H. C. Woerner. Power variation of some integral fractional processes. Bernoulli 12 (2006) 713-735. | Zbl | MR

[5] R. L. Dobrushin and P. Major. Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 (1979) 27-52. | Zbl | MR

[6] L. Giraitis and D. Surgailis. CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 (1985) 191-212. | Zbl | MR

[7] M. Gradinaru and I. Nourdin. Milstein's type scheme for fractional SDEs. Ann. Inst. H. Poincaré Probab. Statist. (2007). To appear. Available at arXiv:math/0702317. | Zbl | MR | Numdam

[8] M. Gradinaru, I. Nourdin, F. Russo and P. Vallois. m-order integrals and Itô's formula for non-semimartingale processes; the case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 781-806. | Zbl | MR | Numdam

[9] M. Gradinaru, F. Russo and P. Vallois. Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index H≥¼. Ann. Probab. 31 (2001) 1772-1820. | Zbl | MR

[10] J. Jacod. Limit of random measures associated with the increments of a Brownian semimartingale. Preprint. Univ. Paris VI (revised version, unpublished work), 1994.

[11] J. León and C. Ludeña. Limits for weighted p-variations and likewise functionals of fractional diffusions with drift. Stochastic Proc. Appl. 117 (2006) 271-296. | Zbl | MR

[12] A. Neuenkirch and I. Nourdin. Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. J. Theoret. Probab. 20 (2007) 871-899. | Zbl | MR

[13] I. Nourdin. A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. In Séminaire de Probabilités XLI 181-197. Springer, Berlin, 2008. | Zbl | MR

[14] I. Nourdin. Asymptotic behavior of some weighted quadratic and cubic variations of the fractional Brownian motion. Ann. Probab. 36 (2008) 2159-2175. | Zbl | MR

[15] I. Nourdin and D. Nualart. Central limit theorems for multiple Skorohod integrals. J. Theoret. Probab. (2008). In revision. Available at arXiv:0707.3448. | Zbl | MR

[16] I. Nourdin and G. Peccati. Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13 (2007) 1229-1256 (electronic). | Zbl | MR

[17] I. Nourdin and A. Réveillac. Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H=1/4. Ann. Probab. (2008). To appear. Available at arXiv:0802.3307. | Zbl | MR

[18] D. Nualart. Malliavin Calculus and Related Topics, 2nd edition. Springer, New York, 2005. | Zbl | MR

[19] D. Nualart. Stochastic calculus with respect to the fractional Brownian motion and applications. Contemp. Math. 336 (2003) 3-39. | Zbl | MR

[20] G. Peccati and C. A. Tudor. Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII 247-262. Lecture Notes in Math. 1857. Springer, Berlin, 2005. | Zbl | MR

[21] M. Taqqu. Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 (1979) 53-83. | Zbl | MR

[22] C. A. Tudor. Analysis of the Rosenblatt process. ESAIM Probab. Statist. 12 230-257. | Zbl | MR | Numdam