Hermite variations of the fractional Brownian sheet

Abstract

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 $0<\alpha \le 1-\frac{1}{2q}$ or $0<\beta \le 1-\frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order $q\ge 2$, 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 $(1,1)$.