{
  "abstract": "Let $Z_n^i=(Z_n^i(1),\\cdots,Z_n^i(d)), n\\geqslant0$, be a supercritical $d$-type branching process starting from a single particle of type $i\\in\\{1,\\cdots,d\\}$, where $d$ is a positive integer. The Kesten-Stigum theorem states that if the mean matrix $M$ of the offspring distribution has spectral radius $ρ>1$ with the associated positive left and right eigenvectors $u$ and $v$, then for all $1\\leqslant i \\leqslant d$, the sequence $W_n^i=\\langle Z_n^i,u\\rangle/ρ^n u(i)$ forms a non-negative martingale, and hence converges almost surely to a random variable $W^i$ as $n\\rightarrow\\infty$. In this paper, we investigate the asymptotic properties of the martingale $\\{W_n^i\\}$, focusing on weighted moments and sharp rates of convergence. We establish necessary and sufficient conditions for the finiteness of $ϕ$-moments of $W^i$, where $ϕ(x)=x^α\\ell(x)$ with $α\\geqslant1$ and $\\ell$ is a positive function slowly varying at infinity. We also determine the sharp exponential rate of convergence of $W_n^i$ to $W^i$ in the $ϕ$-moment sense, identify a phase transition at the critical rate $ρ^{1/2},$ and characterize the behavior at the critical value. Our proofs rely on the double martingale structure of the fundamental martingale and on convex inequalities for martingales.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15273",
  "author_total": 3,
  "authors": [
    {
      "name": "Sheng Xiao"
    },
    {
      "name": "Yingqiu Li"
    },
    {
      "name": "Xingang Liang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eekzyxqwa1cbd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15273",
  "fulltext_url": null,
  "id": "eekzyxqwa1cbd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15273",
  "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",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7221766688",
  "pdf_url": "https://arxiv.org/pdf/2610.15273",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eekzyxqwa1cbd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15273",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15273"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Weighted moments and sharp convergence rates for supercritical multitype branching processes",
  "title_text": "Weighted moments and sharp convergence rates for supercritical multitype branching processes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/eekzyxqwa1cbd",
  "venue": "arXiv",
  "versions": []
}