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Representation and Characterization of Quasistationary Distributions for Markov Chains

Iddo Ben-Ari, Ningwei Jiang

TL;DR

The paper delivers a unified, operator/renewal-based framework for representing and characterizing quasistationary distributions (QSDs) of Markov chains with a unique absorbing state. It separates the analysis into finite and infinite moment-generating-function (MGF) regimes via the critical λ_cr and develops both infinite-regime (Perron-like, minimal QSD ν_cr) and finite-regime (Martin boundary) representations, including concrete existence criteria and explicit formulas. It further extends the theory to continuous time, provides a one-parameter family illustration (hub-and-spokes/tree-like structures), and supplies a suite of examples (skip-free chains, branching processes, rooted trees) to demonstrate existence, uniqueness, and representation. The results connect classical notions like Yaglom limits and R-recurrence to Martin boundary representations, yielding practical tools for constructing and analyzing QSDs in countable state spaces. Overall, the work offers a versatile framework for understanding how QSDs arise, how many can exist under a given λ, and how they can be explicitly represented and approximated in a broad class of Markovian settings.

Abstract

This work provides complete description of Quasistationary Distributions (QSDs) for Markov chains with a unique absorbing state and an irreducible set of non-absorbing states. As is well-known, every QSD has an associated absorption parameter describing the exponential tail of the absorption time under the law of the process with the QSD as the initial distribution. The analysis associated with the existence and representation of QSDs corresponding to a given parameter is according to whether the moment generating function of the absorption time starting from any non-absorbing state evaluated at the parameter is finite or infinite, the finite or infinite moment generating function regimes, respectively. For parameters in the finite regime, it is shown that when exist, all QSDs are in the convex cone of a Martin entry boundary associated with the parameter. The infinite regime corresponds to at most one parameter value and at most one QSD. In this regime, when a QSD exists, it is unique and can be represented by a renewal-type formula. Multiple applications to the findings are presented, including revisiting some of the main classical results in the area.

Representation and Characterization of Quasistationary Distributions for Markov Chains

TL;DR

The paper delivers a unified, operator/renewal-based framework for representing and characterizing quasistationary distributions (QSDs) of Markov chains with a unique absorbing state. It separates the analysis into finite and infinite moment-generating-function (MGF) regimes via the critical λ_cr and develops both infinite-regime (Perron-like, minimal QSD ν_cr) and finite-regime (Martin boundary) representations, including concrete existence criteria and explicit formulas. It further extends the theory to continuous time, provides a one-parameter family illustration (hub-and-spokes/tree-like structures), and supplies a suite of examples (skip-free chains, branching processes, rooted trees) to demonstrate existence, uniqueness, and representation. The results connect classical notions like Yaglom limits and R-recurrence to Martin boundary representations, yielding practical tools for constructing and analyzing QSDs in countable state spaces. Overall, the work offers a versatile framework for understanding how QSDs arise, how many can exist under a given λ, and how they can be explicitly represented and approximated in a broad class of Markovian settings.

Abstract

This work provides complete description of Quasistationary Distributions (QSDs) for Markov chains with a unique absorbing state and an irreducible set of non-absorbing states. As is well-known, every QSD has an associated absorption parameter describing the exponential tail of the absorption time under the law of the process with the QSD as the initial distribution. The analysis associated with the existence and representation of QSDs corresponding to a given parameter is according to whether the moment generating function of the absorption time starting from any non-absorbing state evaluated at the parameter is finite or infinite, the finite or infinite moment generating function regimes, respectively. For parameters in the finite regime, it is shown that when exist, all QSDs are in the convex cone of a Martin entry boundary associated with the parameter. The infinite regime corresponds to at most one parameter value and at most one QSD. In this regime, when a QSD exists, it is unique and can be represented by a renewal-type formula. Multiple applications to the findings are presented, including revisiting some of the main classical results in the area.
Paper Structure (40 sections, 61 theorems, 245 equations, 4 figures)

This paper contains 40 sections, 61 theorems, 245 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Assumptions and Definitions
  3. Prelimiary Results
  4. Results: Discrete-Time
  5. Infinite MGF Regime
  6. Connection with $R$-recurrence
  7. Finite MGF Regime
  8. Martin Boundary Representation
  9. Conditions for Existence of QSDs
  10. Auxiliary Results
  11. Proof of the results of Section \ref{['sec:results_infi_disc']}
  12. Potential Theoretic Results
  13. The Reverse Chain
  14. Proof of Theorem \ref{['th:nu_recurr']}
  15. Proof of Proposition \ref{['prop:K2single']}
  16. ...and 25 more sections

Key Result

Proposition 1.1

Let HD-as:reg:uab,as:reg:Sirr hold. A probability measure $\nu$ on $S$ is a QSD if and only if there exists some $\lambda\in(0,\infty)$ such that In this case,

Figures (4)

  • Figure 1: Cyclic Transfer
  • Figure 2: Regular tree of degree 3 with a unique absorption state $\Delta$
  • Figure 3: Transition Probabilities Diagram
  • Figure 4: Absorption parameter dependence on $\alpha$ for $q=0.95$. The left vertical line corresponds to $r=\sqrt{q(1-q)}$ and the right one to $r\nearrow \rho^{-1} = 4.3589$. Here $e^{\lambda_0}\approx 2.29416$

Theorems & Definitions (118)

  • Definition 1: QSD
  • Proposition 1.1
  • proof
  • Example 1
  • Definition 2
  • Corollary 1.2
  • proof
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • ...and 108 more