We prove functional central and non-central limit theorems for generalized
variations of the anisotropic d-parameter fractional Brownian sheet (fBs) for
any natural number d. Whether the central or the non-central limit theorem
applies depends on the Hermite rank of the variation functional and on the
smallest component of the Hurst parameter vector of the fBs. The limiting
process in the former result is another fBs, independent of the original fBs,
whereas the limit given by the latter result is an Hermite sheet, which is
driven by the same white noise as the original fBs. As an application, we
derive functional limit theorems for power variations of the fBs and discuss
what is a proper way to interpolate them to ensure functional convergence.