Positive Bidiagonal Factorizations for Banded Markov Processes

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 Pi\~neiro and Jacobi-like systems. For Pi\~neiro, the exact PBF region and larger structural-band positivity regions are obtained. Jacobi-like beta-convolution weights admit Gamma-factor cancellations, complete cancellation recovering Pi\~neiro. 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.