{
  "abstract": "We prove two deterministic results for families of distances arising in the study of canonical processes. The first derives an admissible partition scheme from a growth condition. The second gives a representation in terms of parameterized separation trees and compares it with the corresponding majorizing-measure quantities. The main point is that the proofs do not depend on the distribution of the underlying process: once the initial distance and the family of distances are given, no random variables, independence, tail functions, or moment estimates are used. For canonical processes with regular log-concave tails, the assumptions of the abstract results follow from the usual regularity conditions. One direction of the separation-tree estimate also applies to Bernoulli processes without these additional assumptions, and we prove the reverse estimate for bounded convex unconditional index sets. We also give a version of the growth argument for points which, for finite index sets, leads to a recursive construction of admissible partitions and parameterized separation trees.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01709",
  "author_total": 3,
  "authors": [
    {
      "name": "Xuanang Hu"
    },
    {
      "name": "Hanchao Wang"
    },
    {
      "name": "Xinglong Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aha9c0gtsfgn3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01709",
  "fulltext_url": null,
  "id": "aha9c0gtsfgn3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01709",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7215983733",
  "pdf_url": "https://arxiv.org/pdf/2610.01709",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aha9c0gtsfgn3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01709",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01709"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Chaining, tree and measure for some canonical processes",
  "title_text": "Chaining, tree and measure for some canonical processes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/aha9c0gtsfgn3",
  "venue": "arXiv",
  "versions": []
}