Markov Chains Generated by Convolutions of Orthogonality Measures

TL;DR

This paper constructs exactly solvable Markov chains on finite and semi-infinite lattices using convolutions of orthogonality measures from various classical and q-orthogonal polynomials, revealing a property called convolutional self-similarity.

Contribution

It introduces a new class of solvable Markov chains based on convolutions of orthogonality measures, providing explicit stationary distributions and spectral data.

Findings

Stationary distributions exhibit convolutional self-similarity.

Explicit eigenvalues and eigenvectors are derived from orthogonal polynomials.

Constructs include chains related to Krawtchouk, Hahn, Meixner, Charlier, q-Hahn, q-Meixner, and little q-Jacobi polynomials.

Abstract

About two dozens of exactly solvable Markov chains on one-dimensional finite and semi-infinite integer lattices are constructed in terms of convolutions of orthogonality measures of the Krawtchouk, Hahn, Meixner, Charlier, $q$-Hahn, $q$-Meixner and little $q$-Jacobi polynomials. By construction, the stationary probability distributions, the complete sets of eigenvalues and eigenvectors are provided by the polynomials and the orthogonality measures. An interesting property possessed by these stationary probability distributions, called `convolutional self-similarity,' is demonstrated.