Strong approximations for dependent random fields
Abstract
We establish strong Gaussian approximations for centered stationary random fields on ℤd, d≥2, represented as measurable functions of i.i.d.\ innovations. Under finite pth 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 Oa.s.(nmax{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.
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BibTeX
@misc{bonnerjee2026strong,
title = {{Strong approximations for dependent random fields}},
author = {Soham Bonnerjee and Sayar Karmakar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38202},
eprint = {2610.38202},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38202},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Strong approximations for dependent random fields
AU - Soham Bonnerjee
AU - Sayar Karmakar
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38202
UR - https://arxiv.org/abs/2610.38202
AB - We establish strong Gaussian approximations for centered stationary random fields on ℤ^d, d≥2, represented as measurable functions of i.i.d.\ innovations. Under finite pth 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_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.
ER -