{
  "abstract": "The predictable square function measures the size of a martingale. Applied to parameter differences, it also defines a random pseudometric. We explain how to perform a majorizing-measure argument directly in this realized geometry, without conditioning the terminal field to be Gaussian and without first replacing the metric by a random multiple of a deterministic one. For a finite family of predictable Gaussian sums in a $(2,D)$-smooth Banach space, and a fixed probability measure $μ$ on the parameter set, we prove \\[ \\norm{\\max_j\\osc_T f_j}_{L^p} \\le CD\\norm{\\G_μ(d)+\\sqrt p\\,Δ_d}_{L^p},\\qquad p\\ge1, \\] where $Δ_d$ is the diameter of the parameter space in the terminal square-function metric and $\\G_μ(d)$ is its ball-mass integral. The proof combines a localized exponential inequality, pathwise averaging over random balls, and a stopping-time argument. Atomic measures recover logarithmically weighted maximal inequalities; Haar measure connects the result with homogeneous entropy and continuity estimates. We also give extensions, examples, and a precise account of the restrictions on choosing the averaging measure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38425",
  "author_total": 2,
  "authors": [
    {
      "name": "Witold M. Bednorz"
    },
    {
      "name": "Rafał M. Łochowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/et2zz0drg54cf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38425",
  "fulltext_url": null,
  "id": "et2zz0drg54cf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38425",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7215067619",
  "pdf_url": "https://arxiv.org/pdf/2609.38425",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/et2zz0drg54cf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38425",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38425"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Majorizing-measure bounds in the realized square-function metric",
  "title_text": "Majorizing-measure bounds in the realized square-function metric",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/et2zz0drg54cf",
  "venue": "arXiv",
  "versions": []
}