{
  "abstract": "In this work, we develop a unified matrix-level asymptotic calculus for plug-in non-parametric maximum likelihood estimators in finite Markov models. Starting from the asymptotic distribution of the estimated transition matrix, the limiting object is kept in its natural matrix form as a Gaussian random matrix, while the corresponding row-wise vector representation remains immediately available. The main point is that the stochastic constraints of the transition matrix need not be removed by a minimal parametrization: they are carried by the tangent directions and by the covariance structure of the limiting Gaussian matrix, whereas the relevant differentials are computed directly in matrix spaces. A single stochastic calculus theorem gives first-order limit distributions, finite-order developments for sufficiently differentiable functionals, and analytic expansions when the functional is analytic. This provides a common source for asymptotic formulas for matrix powers, stationary characteristics, finite-dimensional curves of Markov characteristics, additive-functional variances, entropy-type quantities and reliability indicators. The resulting covariance operators lead directly to confidence intervals, confidence regions, simultaneous finite-dimensional bands and Wald-type tests. Since the derivations are expressed through matrix products and Kronecker representations rather than coordinate-wise calculations, the method also gives substantial simplifications and, in many cases, computational gains. The second-order terms identify curvature corrections of smooth functionals and provide refined approximations whenever higher-order information is useful.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.17161",
  "author_total": 3,
  "authors": [
    {
      "name": "Georgios Gavrilopoulos"
    },
    {
      "name": "Samis Trevezas"
    },
    {
      "name": "Irene Votsi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0x6gqkadte38w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.17161",
  "fulltext_url": null,
  "id": "0x6gqkadte38w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.17161",
  "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.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7169861778",
  "pdf_url": "https://arxiv.org/pdf/2607.17161",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0x6gqkadte38w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2607.17161",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.17161"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Matrix asymptotic calculus for plug-in maximum likelihood estimators in finite Markov chains",
  "title_text": "Matrix asymptotic calculus for plug-in maximum likelihood estimators in finite Markov chains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:25Z",
  "url": "https://research.draconitic.com/paper/0x6gqkadte38w",
  "venue": "arXiv",
  "versions": []
}