{
  "abstract": "In~\\cite{BFR} we constructed processes $W_n$, built from a family of independent standard Poisson processes, that converge almost surely to the Brownian sheet uniformly on $[0,1]^2$, and in~\\cite{Rov} we proved that $P(\\max_{[0,1]^2}|W_n-W|>αn^{-β})=o(n^{-q})$ for every $β<λ/2$, where $λ\\in(0,\\tfrac{1}{5})$ is the parameter of the construction; this gives rates $n^{-β}$ with $β<\\tfrac{1}{10}$. In this paper, we show that the same construction, with $λ\\geq 1$, satisfies \\[ P\\!\\left(\\max_{(s,t)\\in[0,1]^2} |W_n(s,t)-W(s,t)| > α\\, n^{-1/2}(\\log n)^3\\right) = o(n^{-q}) \\quad\\text{for every }q>0, \\] which is the one-parameter rate of Gorostiza and Griego~\\cite{GG} up to a factor $(\\log n)^{1/2}$. In particular the restriction $λ<\\tfrac{1}{5}$ is not needed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23347",
  "author_total": 1,
  "authors": [
    {
      "name": "Carles Rovira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/10e89nrapwkvq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23347",
  "fulltext_url": null,
  "id": "10e89nrapwkvq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23347",
  "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"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7222972959",
  "pdf_url": "https://arxiv.org/pdf/2610.23347",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/10e89nrapwkvq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23347",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23347"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Uniform transport processes and the Brownian sheet:the one-parameter rate of convergence is preserved",
  "title_text": "Uniform transport processes and the Brownian sheet:the one-parameter rate of convergence is preserved",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/10e89nrapwkvq",
  "venue": "arXiv",
  "versions": []
}