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Quasi-stationary distributions in reducible state spaces

Nicolas Champagnat, Denis Villemonais

TL;DR

The paper addresses the existence and precise asymptotics of quasi-stationary distributions for absorbed Markov chains on reducible state spaces by introducing exponential and polynomial convergence parameters $\theta_{0,S}$ and $j_S$, together with Assumption (A) that decomposes the space into three interacting parts. Using a triangular-operator framework, it derives explicit leading-term representations for the semigroup and convergence rates, and fully characterizes the quasi-stationary distributions as convex combinations of class-specific measures $\nu_{S,i}$ with polynomial corrections governed by $j_S$; it provides a detailed analysis for two-set reductions (sources, sinks, and critical sinks) and extends to multiple communication classes. The results yield existence and explicit structure of QSDs in finite and countable class settings, with a constructive description of convergence speeds to quasi-limiting laws. Finally, the paper proves that discrete-state Markov chains on denumerable spaces admit QSDs under mild Lyapunov and aperiodicity assumptions, without irreducibility, broadening the applicability of QSD theory to general reducible systems.

Abstract

We study quasi-stationary distributions and quasi-limiting behavior of Markov chains in general reducible state spaces with absorption. We propose a set of assumptions dealing with particular situations where the state space can be decomposed into three subsets between which communication is only possible in a single direction. These assumptions allow us to characterize the exponential order of magnitude and the exact polynomial correction, called polynomial convergence parameter, for the leading order term of the semigroup for large time. They also provide explicit convergence speeds to this leading order term. We apply these results to general Markov chains with finitely or denumerably many communication classes using a specific induction over the communication classes of the chain. We are able to explicitely characterize the polynomial convergence parameter, to determine the complete set of quasistationary distributions and to provide explicit estimates for the speed of convergence to quasi-limiting distributions in the case of finitely many communication classes. We conclude with an application of these results to the case of denumerable state spaces, where we are able to prove that, in general, there is existence of a quasi-stationary distribution without assuming irreducibility before absorption. This actually holds true assuming only aperiodicity, the existence of a Lyapunov function and the existence of a point in the state space from which the return time is finite with positive probability.

Quasi-stationary distributions in reducible state spaces

TL;DR

The paper addresses the existence and precise asymptotics of quasi-stationary distributions for absorbed Markov chains on reducible state spaces by introducing exponential and polynomial convergence parameters and , together with Assumption (A) that decomposes the space into three interacting parts. Using a triangular-operator framework, it derives explicit leading-term representations for the semigroup and convergence rates, and fully characterizes the quasi-stationary distributions as convex combinations of class-specific measures with polynomial corrections governed by ; it provides a detailed analysis for two-set reductions (sources, sinks, and critical sinks) and extends to multiple communication classes. The results yield existence and explicit structure of QSDs in finite and countable class settings, with a constructive description of convergence speeds to quasi-limiting laws. Finally, the paper proves that discrete-state Markov chains on denumerable spaces admit QSDs under mild Lyapunov and aperiodicity assumptions, without irreducibility, broadening the applicability of QSD theory to general reducible systems.

Abstract

We study quasi-stationary distributions and quasi-limiting behavior of Markov chains in general reducible state spaces with absorption. We propose a set of assumptions dealing with particular situations where the state space can be decomposed into three subsets between which communication is only possible in a single direction. These assumptions allow us to characterize the exponential order of magnitude and the exact polynomial correction, called polynomial convergence parameter, for the leading order term of the semigroup for large time. They also provide explicit convergence speeds to this leading order term. We apply these results to general Markov chains with finitely or denumerably many communication classes using a specific induction over the communication classes of the chain. We are able to explicitely characterize the polynomial convergence parameter, to determine the complete set of quasistationary distributions and to provide explicit estimates for the speed of convergence to quasi-limiting distributions in the case of finitely many communication classes. We conclude with an application of these results to the case of denumerable state spaces, where we are able to prove that, in general, there is existence of a quasi-stationary distribution without assuming irreducibility before absorption. This actually holds true assuming only aperiodicity, the existence of a Lyapunov function and the existence of a point in the state space from which the return time is finite with positive probability.
Paper Structure (9 sections, 15 theorems, 199 equations, 1 figure)

This paper contains 9 sections, 15 theorems, 199 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Exponential and polynomial convergence parameter
  3. Quasi-stationary distributions in reducible state spaces with two successive sets
  4. Polynomial decay for upper triangular matrix of linear operators
  5. Case where $D_1$ is a source ($\theta_{0,P}>\theta_{0,R}$)
  6. Case where $D_1$ is a sink ($\theta_{0,P}<\theta_{0,R}$)
  7. Case where $D_1$ is a critical sink ($\theta_{0,P}=\theta_{0,R}$)
  8. Reducible state spaces with several communication classes
  9. Discrete state spaces

Key Result

Proposition 2.1

For all $\mu\in\mathcal{M}_+(D)$, and If Assumption (A) holds true, then $j_S$ is lower semi-continuous on $\mathcal{M}_+(W_S)$ and, for all $\mu\in \mathcal{M}_+(W_S)$, and In addition, and $(j_S(X_n))_{n\geq 0}$ is $\mathbb{P}_x$-almost surely non-increasing, for all $x\in D$.

Figures (1)

  • Figure 1: Transition graph displaying the relations between the sets $D_1$, $D_2$ and $\partial$. The dashed lines indicate the domains and co-domains of the sub-Markov kernels $P,Q,R$.

Theorems & Definitions (44)

  • Remark 1
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['prop:propajs1']}
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Remark 2
  • proof
  • ...and 34 more