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

Amílcar Branquinho, Ana Foulquié-Moreno, Manuel Mañas

TL;DR

The paper addresses discrete-time Markov chains with bounded banded transition matrices that admit a positive bidiagonal factorization, extending beyond the classical birth–death setting. It leverages a spectral Favard framework to produce a canonical stochastic factorization through diagonal conjugations and derives Karlin--McGregor type representations using mixed-type multiple orthogonal polynomials and matrix-valued spectral measures. In the finite case, it yields explicit formulas for transition probabilities, first-passage generating functions, recurrence, and stationary distributions with geometric convergence, while in the countably infinite case it provides analogous spectral representations and precise recurrence/ergodicity criteria via the spectral measure, including a mass-at-one condition for positive recurrence. The results unify and extend previous Hessenberg/tetradiagonal analyses within a robust banded-spectral framework and point to extensions to unbounded bands, continuous-time processes, and perturbations, highlighting deep connections with total positivity and integrable structures.

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.

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

TL;DR

The paper addresses discrete-time Markov chains with bounded banded transition matrices that admit a positive bidiagonal factorization, extending beyond the classical birth–death setting. It leverages a spectral Favard framework to produce a canonical stochastic factorization through diagonal conjugations and derives Karlin--McGregor type representations using mixed-type multiple orthogonal polynomials and matrix-valued spectral measures. In the finite case, it yields explicit formulas for transition probabilities, first-passage generating functions, recurrence, and stationary distributions with geometric convergence, while in the countably infinite case it provides analogous spectral representations and precise recurrence/ergodicity criteria via the spectral measure, including a mass-at-one condition for positive recurrence. The results unify and extend previous Hessenberg/tetradiagonal analyses within a robust banded-spectral framework and point to extensions to unbounded bands, continuous-time processes, and perturbations, highlighting deep connections with total positivity and integrable structures.

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 in the spectral measure, corresponding to the eigenvalue with both right and left eigenvectors.
Paper Structure (7 sections, 21 theorems, 93 equations)

This paper contains 7 sections, 21 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Elements of Markov chains
  3. Markov chains with banded transition matrices
  4. Finite Markov chains
  5. Countable infinite Markov chains
  6. Totally nonnegative and oscillatory matrices
  7. Recursion relations and their consequences

Key Result

Theorem 1.8

The states in the same class are either all recurrent or all transient and have the same period.

Theorems & Definitions (56)

  • Definition 1.1: Period and aperiodic states
  • Remark 1.2
  • Definition 1.3: First-passage-time probability
  • Definition 1.4: Recurrence and transience
  • Remark 1.5
  • Definition 1.6: Ergodicity
  • Definition 1.7: Classes of states
  • Theorem 1.8: Class properties
  • Definition 1.9: Irreducible Markov chain
  • Definition 1.10: Expected return times
  • ...and 46 more