{
  "abstract": "We give an analytic proof of the comparison conjectured by Das, Łochowski, Matsuda and Perkowski for the fractional Brownian level-crossing constant: for standard fractional Brownian motion, $c_H>\\mathbb{E}|Z|^{1/H}$ when $0<H<1/2$ and $c_H<\\mathbb{E}|Z|^{1/H}$ when $1/2<H<1$, where $Z\\sim N(0,1)$. At $H=1/2$, both constants equal one. The main tool is Gaussian comparison under ordering of increment variances. It applies because truncated variation is translation invariant and lattice-submodular, as follows from its minimum-total-variation characterization. A tangent variogram produces Brownian motion with an independent Gaussian drift on the upper side, and a periodic Brownian bridge on the lower side. The former has an explicit stationary reflection rate. For the latter we calculate an integrated cyclic crossing count by first-passage Laplace transforms. Elementary hyperbolic, theta-series, and Gamma inequalities then give strict comparisons on both full parameter ranges. No numerical integration, simulation, interval certificate, or quantitative local expansion at $H=1/2$ is used.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38894",
  "author_total": 2,
  "authors": [
    {
      "name": "W. M. Bednorz"
    },
    {
      "name": "R. M. Łochowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xh4sa9we1gx6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38894",
  "fulltext_url": null,
  "id": "0xh4sa9we1gx6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38894",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38894",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xh4sa9we1gx6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38894",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38894"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Proof of the conjecture about inequality of $1/H$-variations of fractional Brownian motion along the equidistant and the Lebesgue partitions",
  "title_text": "Proof of the conjecture about inequality of 1/H-variations of fractional Brownian motion along the equidistant and the Lebesgue partitions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/0xh4sa9we1gx6",
  "venue": "arXiv",
  "versions": []
}