{
  "abstract": "When the density at the antipode of the intrinsic Fréchet mean reaches a critical level, an interval for the mean of a long-memory circular time series that is calibrated to independent-data scaling is too short, and at Hermite rank two, it also has the wrong shape. The antipode is the cut locus, where squared distance has a kink, and the quadratic coefficient of the population objective vanishes exactly when the density there equals 1/(2π). For Yt=G(Zt), subordinating a stationary Gaussian sequence with covariances decaying at rate α∈(0,1), and contact of order 2r at the antipode, the rescaled estimator converges to a signed (2r+1)-st root. The contact order fixes the root and the Hermite rank of the score fixes what sits under it: a Gaussian at rank one, or a Rosenblatt variable at rank two under strong memory. Because the critical density is a knife-edge equality, we also give the limit along an array approaching it: when the quadratic coefficient vanishes at a rate matched to the long-memory scale of the score, the limit is a one-parameter family joining the odd root to a linear one. In hourly wind directions from fifty-three stations, the antipodal density ranges from 0.05 to 0.74 of the critical level; the records identify no subordination and do not settle the memory, so they locate series relative to the critical level rather than testing the limits.",
  "abstract_shown": true,
  "affiliations": [
    "School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China",
    "School of Economics, Beijing Institute of Technology, Beijing 100081, China"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Hongxuan Yan",
      "orcid": "0000-0002-1882-7685"
    },
    {
      "affs": [
        1
      ],
      "name": "Luoyi Sun",
      "orcid": "0009-0004-6916-393X"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8bz3cn158smsf.bib",
  "cited_by": 1,
  "doi": "10.3390/fractalfract10100725",
  "fulltext_url": null,
  "id": "8bz3cn158smsf",
  "issns": [
    "2504-3110"
  ],
  "landing_url": "https://www.mdpi.com/2504-3110/10/10/725",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2002|Complex Systems and Time Series Analysis"
    ]
  ],
  "openalex": "W7222980960",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/8bz3cn158smsf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:32Z",
      "native_id": "10.3390/fractalfract10100725",
      "source": "crossref",
      "url": "https://www.mdpi.com/2504-3110/10/10/725"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2504-3110",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Signed Odd-Root Limits for Intrinsic Fréchet Means on the Circle Under Long-Memory Gaussian Subordination",
  "title_text": "Signed Odd-Root Limits for Intrinsic Fréchet Means on the Circle Under Long-Memory Gaussian Subordination",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:32Z",
  "url": "https://research.draconitic.com/paper/8bz3cn158smsf",
  "venue": "Fractal and Fractional",
  "versions": []
}