Hermite variations of the fractional Brownian sheet

We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet $W^{α, β}$ with Hurst parameter $(α, β) \in (0,1)^2$. When $0<α\leq 1-\frac{1}{2q}$ or $0<β\leq 1-\frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order $q\geq 2$, while for $1-\frac{1}{2q}<α, β< 1$ 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 $(1,1)$.

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