Positive Bidiagonal Factorizations for Banded Markov Processes
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× 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∈{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.
Related
- Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials
- Positive Bidiagonal Factorizations: Combinatorial Structure, Spectral Theory, and Matrix Continued Fractions
- Exact Factorizations of Rank-One Pointed Hopf Algebras in Positive Characteristic, I
- Accumulated-Variance Functionals of Hidden Markov and Semi-Markov Processes: Finite-Sample Upper and Lower Bounds
- Worst-Case Bayesian CVaR Regret Bounds for Episodic Markov Decision Processes
BibTeX
@misc{manas2026positive,
title = {{Positive Bidiagonal Factorizations for Banded Markov Processes}},
author = {Manuel Mañas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.00788},
eprint = {2608.00788},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00788},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive Bidiagonal Factorizations for Banded Markov Processes
AU - Manuel Mañas
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.00788
UR - https://arxiv.org/abs/2608.00788
AB - 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× 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∈{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.
ER -