Strong approximations for dependent random fields

Abstract

We establish strong Gaussian approximations for centered stationary random fields on $\mathbb Z^d$, $d\ge2$, represented as measurable functions of i.i.d.\ innovations. Under finite $p$th moments, $p>2$, and sufficiently fast decay of spatial dependence, we construct a single coupling of the field with an i.i.d.\ Gaussian field. Its partial sums, scaled by the long-run standard deviation, approximate those of the original field almost surely, uniformly over all rectangular prefixes and all observation scales. Over a cube of side length $n$, the approximation error is $O_{\mathrm{a.s.}}(n^{\max\{d/p,(d-1)/2\}})$ up to logarithmic factors. While $d/p$ is the familiar moment term, the surface term $(d-1)/2$, absent for independent planar arrays, emerges as a cost of dependence, and cannot be removed even for finite-range Gaussian fields. The assumptions accommodate nonlinear and noncausal fields without imposing mixing. The proof combines a permanent dyadic blocking, a local Gaussian coupling in tensor Haar coordinates driven by a Doob martingale with vanishing conditional third moments, and a comparison of the resulting dependent Gaussian blocks with white noise.