  We establish a central limit theorem for partial sums of stationary linear
random fields with dependent innovations, and an invariance principle for
anisotropic fractional Brownian sheets. Our result is a generalization of the
invariance principle for fractional Brownian motions by Dedecker et al. (2011)
to high dimensions. A key ingredient of their argument, the martingale
approximation, is replaced by an m-approximation argument. An important tool of
our approach is a moment inequality for stationary random fields recently
established by El Machkouri et al. (2011).
