{
  "abstract": "The branching ratio $ρ$ of a self-exciting point process determines the branching margin $1-ρ$ to the critical boundary. We study inference on $ρ$ from a finite record for stationary Hawkes processes with completely monotone kernels whose slow relaxation mass obeys an envelope near zero rate. Slow excitation can then be exchanged for exogenous immigration; a spectral bound on the Kullback--Leibler rate and a coupling argument for the unobserved past give a minimax lower bound of order $T^{-γ/(2γ+1)}$ for a power-law envelope with tail exponent $γ$. A block-dispersion estimator whose block length grows as $T^{1/(2γ+1)}$ attains this rate. A quantitative central limit theorem, obtained from the cluster representation and a second-order Poincaré inequality on Poisson space, yields honest confidence intervals of nearly minimax expected length. For unknown $γ$, a Lepski-type rule adapts in probability at a logarithmic cost; free adaptation between memory classes is impossible, and honest intervals cannot adapt to lighter memory. Simulations are consistent with the rates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38267",
  "author_total": 1,
  "authors": [
    {
      "name": "Mauricio Herrera-Marín"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cy2fv1sqt4w29.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38267",
  "fulltext_url": null,
  "id": "cy2fv1sqt4w29",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38267",
  "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.AP"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7215055061",
  "pdf_url": "https://arxiv.org/pdf/2609.38267",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cy2fv1sqt4w29.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38267",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38267"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Minimax and adaptive inference for branching ratios under long memory",
  "title_text": "Minimax and adaptive inference for branching ratios under long memory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:58Z",
  "url": "https://research.draconitic.com/paper/cy2fv1sqt4w29",
  "venue": "arXiv",
  "versions": []
}