Spectral theory for Markov chains with transition matrix admitting a stochastic bidiagonal factorization

TL;DR

This paper extends spectral analysis techniques to a broader class of Markov chains with banded transition matrices that admit a positive stochastic bidiagonal factorization, deriving spectral representations and stationary distributions.

Contribution

It generalizes the spectral Favard theorem to Markov chains beyond birth-and-death processes, providing explicit formulas and characterizations for recurrence, stationary distributions, and ergodicity.

Findings

Derived Karlin-McGregor spectral representation for finite case.

Established recurrence and explicit stationary distributions.

Characterized behavior and ergodicity in the infinite case.

Abstract

The recently established spectral Favard theorem for bounded banded matrices admitting a positive bidiagonal factorization is applied to a broader class of Markov chains with bounded banded transition matrices, extending beyond the classical birth-and-death setting, to those that allow a positive stochastic bidiagonal factorization. In the finite case, the Karlin-McGregor spectral representation is derived. The recurrence of the Markov chain is established, and explicit formulas for the stationary distributions are provided in terms of orthogonal polynomials. Analogous results are obtained for the countably infinite case. In this setting, the chain is not necessarily recurrent, and its behavior is characterized in terms of the associated spectral measure. Finally, ergodicity is examined through the presence of a mass at $1$ in the spectral measure, corresponding to the eigenvalue $1$ with both right and left eigenvectors.