Hermite variations of the fractional Brownian sheet

Anthony Réveillac (1) , Michael Stauch (1) , Ciprian A. Tudor (2)

1 Institut für Mathematik [Berlin]
2 LPP - Laboratoire Paul Painlevé - UMR 8524

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 $(1,1)$.

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

Domaines: Probabilités [math.PR]

Anthony Réveillac, Michael Stauch, Ciprian A. Tudor. Hermite variations of the fractional Brownian sheet. 2010. ⟨hal-00522801⟩