Finite Markov chains and multiple orthogonal polynomials

TL;DR

This paper studies finite Markov chains derived from orthogonal and multiple orthogonal polynomials, analyzing their spectral properties, recurrence behaviors, and reversibility, with explicit examples from hypergeometric families within the Askey scheme.

Contribution

It introduces a comprehensive spectral analysis framework for finite Markov chains constructed via orthogonal polynomials, including multiple orthogonal polynomials, and provides explicit cases within the Askey scheme.

Findings

Recurrent reversible birth and death chains identified within the Askey scheme.

Explicit spectral properties and stationary states derived for various orthogonal polynomial families.

Factorization into pure birth and pure death processes demonstrated for these chains.

Abstract

This paper investigates stochastic finite matrices and the corresponding finite Markov chains constructed using recurrence matrices for general families of orthogonal polynomials and multiple orthogonal polynomials. The paper explores the spectral theory of transition matrices, utilizing both orthogonal and multiple orthogonal polynomials. Several properties are derived, including classes, periodicity, recurrence, stationary states, ergodicity, expected recurrence times, time-reversed chains, and reversibility. Furthermore, the paper uncovers factorization in terms of pure birth and pure death processes. The case study focuses on hypergeometric orthogonal polynomials, where all the computations can be carried out effectively. Particularly within the Askey scheme, all descendants under Hahn (excluding Bessel), such as Hahn, Jacobi, Meixner, Kravchuk, Laguerre, Charlier, and Hermite, present interesting examples of recurrent reversible birth and death finite Markov chains. Additionally, the paper considers multiple orthogonal polynomials, including multiple Hahn, Jacobi-Pi\~neiro, Laguerre of the first kind, and Meixner of the second kind, along with their hypergeometric representations and derives the corresponding recurrent finite Markov chains and time-reversed chains.