{
  "abstract": "We record two observations concerning \\cite{HLW24}. First, the proof of its Theorem~1 yields the population linearization (13)--(14) at \\textit{any critical point} of the Fréchet function $F_μ$, under its remaining geometric and measure-theoretic assumptions; neither isolation nor a nonsingular Hessian of $F_μ$ is needed. Second, its Theorem~2 is given a shorter proof by applying the M-estimator theorem \\cite[Theorem~5.23, pp.~53--54]{vdV98} to negative squared distance in normal coordinates, as opposed to the Donsker argument used there. We verify the pointwise differentiability, local square-integrable Lipschitz bound, second order expansion of population loss, consistency, and empirical maximization hypotheses. The resulting covariance retains the cut-locus correction. The same argument applies to suitably selected local sample Fréchet minima when the population Hessian is positive definite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07956",
  "author_total": 1,
  "authors": [
    {
      "name": "Susovan Pal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7p4jer416bf9h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07956",
  "fulltext_url": null,
  "id": "7p4jer416bf9h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07956",
  "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.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07956",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7p4jer416bf9h.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07956",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07956"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A Note on the Cut-Locus-Corrected Fréchet Mean Central Limit Theorem",
  "title_text": "A Note on the Cut-Locus-Corrected Fréchet Mean Central Limit Theorem",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:52Z",
  "url": "https://research.draconitic.com/paper/7p4jer416bf9h",
  "venue": "arXiv",
  "versions": []
}