{
  "abstract": "Khaleghi and Lugosi asked whether the sum of the $ϕ$-mixing (uniform-mixing) coefficients of a real-valued discrete-time stationary ergodic process can be consistently estimated from a single sample path. We construct a deterministic sequence of Borel statistics that, for every such process, converges almost surely to the extended sum $\\sum_{m\\geq1}ϕ(m)$, including divergence to $+\\infty$ when the sum is infinite. The estimator combines finite dyadic cylinders with a vanishing empirical cutoff on conditioning-event frequencies. When the target is finite, $α(m)\\leqϕ(m)$ supplies summable covariance control for the growing finite classes, and a stable-division estimate yields the upper bound. Fixed positive-probability witnesses and Birkhoff's theorem give the lower bound, including the infinite-target case. No mixing rate or known positive lower bound on conditioning-event probabilities is required.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26232",
  "author_total": 1,
  "authors": [
    {
      "name": "Senhan Yao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0qna4xq6mv6bk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26232",
  "fulltext_url": null,
  "id": "0qna4xq6mv6bk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26232",
  "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.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7214067248",
  "pdf_url": "https://arxiv.org/pdf/2609.26232",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0qna4xq6mv6bk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26232",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26232"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Strongly Consistent Estimation of the Extended $\\ell_1$-Sum of $ϕ$-Mixing Coefficients from a Single Trajectory",
  "title_text": "Strongly Consistent Estimation of the Extended ℓ_1-Sum of ϕ-Mixing Coefficients from a Single Trajectory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:49Z",
  "url": "https://research.draconitic.com/paper/0qna4xq6mv6bk",
  "venue": "arXiv",
  "versions": []
}