论文标题
振荡性带的黑森贝格矩阵，多个正交多项式和随机步行
Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and random walks
论文作者
论文摘要
发现了一个接收阳性比二角分解的有限带的下赫森伯格矩阵的光谱定理。关于振荡矩阵的光谱和分解特性的广泛知识，就一组正交I和II的多种正交多项式的序列而言，相对于一组阳性Lebesgue-Stieltjes〜措施，该频谱最爱。还证明了多个高斯正交正交正交正交，并发现了相应的精度。 该频谱最佳定理适用于$（p+2）$ - 对角线过渡矩阵，即超出生育和死亡的马尔可夫链，这些矩阵承认了积极的随机双节性分解。在有限情况下，给出了Karlin-McGregor光谱表示。结果表明，马尔可夫链是反复出现的，并且根据固定分布的正交多项式而言，明确表达式。对于可数的无限马尔可夫链也获得了类似的结果。现在，马尔可夫链不一定经常出现，它的特征是第一个措施。根据$ 1 $的质量的存在，讨论了马尔可夫链的奇特性，这是与左右特征向量相对应的特征值。
A spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization is found. The large knowledge on the spectral and factorization properties of oscillatory matrices leads to this spectral Favard theorem in terms of sequences of multiple orthogonal polynomials of types I and II with respect to a set of positive Lebesgue-Stieltjes~measures. Also a multiple Gauss quadrature is proven and corresponding degrees of precision are found. This spectral Favard theorem is applied to Markov chains with $(p+2)$-diagonal transition matrices, i.e. beyond birth and death, that admit a positive stochastic bidiagonal factorization. In the finite case, the Karlin-McGregor spectral representation is given. It is shown that the Markov chains are recurrent and explicit expressions in terms of the orthogonal polynomials for the stationary distributions are given. Similar results are obtained for the countable infinite Markov chain. Now the Markov chain is not necessarily recurrent, and it is characterized in terms of the first measure. Ergodicity of the Markov chain is discussed in terms of the existence of a mass at $1$, which is an eigenvalue corresponding to the right and left eigenvectors.
