{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38202",
  "author_total": 2,
  "authors": [
    {
      "name": "Soham Bonnerjee"
    },
    {
      "name": "Sayar Karmakar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9jx3d40tqc1kv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38202",
  "fulltext_url": null,
  "id": "9jx3d40tqc1kv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38202",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9jx3d40tqc1kv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Strong approximations for dependent random fields",
  "title_text": "Strong approximations for dependent random fields",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:00Z",
  "url": "https://research.draconitic.com/paper/9jx3d40tqc1kv",
  "venue": "arXiv",
  "versions": []
}