Hermite variations of the fractional Brownian sheet

Réveillac, Anthony; Stauch, Michael; Tudor, Ciprian A.

Réveillac, Anthony; Stauch, Michael; Tudor, Ciprian A. (2012), Hermite variations of the fractional Brownian sheet, Stochastics and Dynamics, 12, 3. http://dx.doi.org/10.1142/S0219493711500213

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00522801/fr/

Date

2012

Journal name

Stochastics and Dynamics

Volume

Number

Publisher

World Scientific

Abstract (EN)

We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet $W^{\alpha, \beta}$ with Hurst parameter $(\alpha, \beta) \in (0,1)^2$. When $0<\alpha \leq 1-\frac{1}{2q}$ or $0<\beta \leq 1-\frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order $q\geq 2$, while for $1-\frac{1}{2q}<\alpha, \beta < 1$ we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time $(1,1)$.

Subjects / Keywords

Limit theorems; Hermite variations; Multiple stochastic integrals; Malliavin calculus; Weak convergence