{
  "abstract": "Let $Z_{i,n}$ be the population at time $n$ descended from one particle at time $i$ in a branching process in a varying environment. Under two-sided bounds on products of offspring means, an integrable uniform second-moment tail condition, and nondegeneracy and unit positive-support gcd of all weak limits, we prove $\\mathbb{P}(Z_{i,n}=k)\\le C(n-i+1)^{-2}$ uniformly in $0\\le i<n$ and $k\\ge1$, and obtain a local limit theorem as $n-i,k\\to\\infty$ with $k\\le L_0(n-i)$, without third-moment assumptions. These results extend the classical local probability estimates for critical Galton--Watson processes to varying environments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01455",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhiyang Xia"
    },
    {
      "name": "Mei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/41egrkbdychzd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01455",
  "fulltext_url": null,
  "id": "41egrkbdychzd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01455",
  "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.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01455",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/41egrkbdychzd.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01455",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01455"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Local Limit Theorem for Branching Processes in Varying Environments",
  "title_text": "Local Limit Theorem for Branching Processes in Varying Environments",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/41egrkbdychzd",
  "venue": "arXiv",
  "versions": []
}