Markov Chains and Multiple Orthogonality

TL;DR

This paper explores the relationship between Markov chains and multiple orthogonality, providing a method to generate specific stochastic Hessenberg matrices linked to multiple orthogonal polynomials.

Contribution

It introduces a procedure to generate stochastic tetra diagonal Hessenberg matrices from families of multiple orthogonal polynomials, expanding the understanding of their structure and interrelations.

Findings

Constructed stochastic tetra diagonal Hessenberg matrices from multiple orthogonal polynomials.

Showed that associated matrices are limit transposes of each other.

Connected the matrices' properties with the Poincaré theorem.

Abstract

In this work we survey on connections of Markov chains and the theory of multiple orthogonality. Here we mainly concentrate on give a procedure to generate stochastic tetra diagonal Hessenberg matrices, coming from some specific families of multiple orthogonal, such as the ones of Jacobi--Pi\~neiro and Hypergeometric Lima--Loureiro. We show that associated with a positive tetra diagonal nonnegative bounded Hessenberg matrix we can construct two stochastic tetra diagonal ones. These two stochastic tridiagonal nonnegative Hessenberg matrices are shown to be, enlightened by the Poincar\'e theorem, limit transpose of each other.