{
  "abstract": "An ordered positive bidiagonal factorization (PBF) is used to develop a spectral and probabilistic theory for Markov transition matrices of arbitrary finite bandwidth. The factors determine two families of mixed-type multiple orthogonal polynomials, an entrywise positive $q\\times p$ matrix of measures, and a sequence of elementary death-or-stay and birth-or-stay transitions. This yields Karlin-McGregor formulas for transition probabilities, Green kernels, resolvents, potentials, and first-passage transforms without reversibility or block symmetrizability. Rational stochastic PBFs are characterized by ordered finite-urn experiments. Cyclic reorderings give Darboux intertwinings, while a factor-resolved continued fraction gives the first-return law. Grouping states produces a finite-phase quasi-birth-and-death process and a matrix continued fraction for return time and phase. For bounded continuous-time generators, uniformization preserves the spectral data. For unbounded rates, a conservative generator whose shifted leading truncations all admit scalar PBFs must be tridiagonal; nevertheless, $Q=V(T-I)$ preserves arbitrary fixed bandwidth and separates the embedded chain from the holding rates. The theory is explicit for mixed Piñeiro and Jacobi-like systems. For Piñeiro, the exact PBF region and larger structural-band positivity regions are obtained. Jacobi-like beta-convolution weights admit Gamma-factor cancellations, complete cancellation recovering Piñeiro. For $q\\in\\{2,3,4\\}$, the strict ordering conditions give the only open PBF region, with further lower-dimensional cancellation strata. Rational $(3,2)$ examples provide all factors and the resulting Markov models.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.00788",
  "author_total": 1,
  "authors": [
    {
      "name": "Manuel Mañas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1hte20c2zr1eg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.00788",
  "fulltext_url": null,
  "id": "1hte20c2zr1eg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.00788",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
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      "arxiv",
      "math.PR"
    ],
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      "arxiv",
      "math.SP"
    ],
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      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
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  "openalex": "W7172448136",
  "pdf_url": "https://arxiv.org/pdf/2608.00788",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1hte20c2zr1eg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:52Z",
      "native_id": "2608.00788",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.00788"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Positive Bidiagonal Factorizations for Banded Markov Processes",
  "title_text": "Positive Bidiagonal Factorizations for Banded Markov Processes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/1hte20c2zr1eg",
  "venue": "arXiv",
  "versions": []
}