Invariance principles for non-isotropic long memory random fields

We prove that when a random field with bounded spectral density satisfies a Donsker type theorem, its dilated and properly normalised spectral field admits a weak limit. We apply this result to establish the convergence of partial sums for random fields obtained by filtering a white noise. In particular, we prove the convergence of partial sums for strongly-dependent fields whose memory does not satisfy the regularity conditions previously met in the literature.