Finite Markov chains and multiple orthogonal polynomials

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, using 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 representations of orthogonal polynomials, where all the computations can be carried out effectively. Particularly within the Askey scheme, all descendants under Hahn such as Hahn itself, 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–Piñeiro, 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. A Mathematica code, publicly accessible in repositories, has been crafted to analyze various features within finite Markov chains.