发表机构
Complutense University of Madrid（马德里康普顿斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出正双对角分解（PBF）理论，建立无可逆性等条件下带状马尔可夫过程的谱与概率框架，推导相关公式并刻画其结构，给出显式示例。
AI 中文摘要
有序正双对角分解（PBF）被用于建立任意有限带宽马尔可夫转移矩阵的谱理论与概率理论。这些因子确定两类混合型多重正交多项式、一个逐元素正的q×p测度矩阵，以及一系列基本的“死亡或保留”与“出生或保留”转移。这得到了无可逆性或块对称可分性条件下的转移概率、格林核、预解式、位势及首通变换的Karlin-McGregor公式。有理随机PBF由有序有限瓮实验刻画；循环重排给出Darboux互缠，而因子分解的连分数给出首回返律。分组状态产生有限相位拟生灭过程及回返时间与相位的矩阵连分数。对于有界连续时间生成元，均匀化保留谱数据；对于无界速率，若移位后的前向截断均允许标量PBF的保守生成元必为三对角，但Q=V(T-I)保留任意固定带宽，并将嵌入链与停留速率分离。该理论对混合型Piñeiro系统及类Jacobi系统是显式的：对Piñeiro系统，得到精确PBF区域及更大的结构带正性区域；类Jacobi beta卷积权重允许Gamma因子抵消，完全抵消后恢复Piñeiro系统；当q∈{2,3,4}时，严格有序条件给出唯一开放的PBF区域，另有更低维的抵消层；有理(3,2)示例提供所有因子及对应的马尔可夫模型。
英文摘要
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.
