论文标题
准生气和死亡过程和多元正交多项式
Quasi-birth-and-death processes and multivariate orthogonal polynomials
论文作者
论文摘要
本文的目的是研究由双变量正交多项式理论引起的一些准生物和死亡（QBD）过程的模型。首先，我们将看到如何在一般环境中执行光谱分析，以及根据支持某些域$ω\ subset \ subset \ mathbb {r}^d $的光谱度量的复发和这些过程的不变度度量的结果。之后，我们将将结果应用于双变量正交多项式的几个示例，即产物正交多项式，在抛物线域上的正交多项式和三角形上的正交多项式。我们将重点关注这些多项式产生的jacobi矩阵的线性组合，并产生连续或离散时间QBD过程的家族。最后，我们显示了一些与这些QBD过程相关的URN模型。
The aim of this paper is to study some models of quasi-birth-and-death (QBD) processes arising from the theory of bivariate orthogonal polynomials. First we will see how to perform the spectral analysis in the general setting as well as to obtain results about recurrence and the invariant measure of these processes in terms of the spectral measure supported on some domain $Ω\subset\mathbb{R}^d$. Afterwards, we will apply our results to several examples of bivariate orthogonal polynomials, namely product orthogonal polynomials, orthogonal polynomials on a parabolic domain and orthogonal polynomials on the triangle. We will focus on linear combinations of the Jacobi matrices generated by these polynomials and produce families of either continuous or discrete-time QBD processes. Finally, we show some urn models associated with these QBD processes.
