Quasi-stationary distributions of non-absorbing Markov chains

TL;DR

This paper introduces a new concept of quasi-stationary distribution for reversible ergodic Markov chains without absorbing states, generalizing classical notions and providing geometric and spectral insights.

Contribution

It defines a novel quasi-stationary distribution based on strong stationary times, extending the classical theory to non-absorbing chains and exploring its properties and limitations.

Findings

The new distribution generalizes the Yaglom limit.

It can be expressed via eigenvectors of the Markov kernel.

The phenomenology is richer than in the absorbing case.

Abstract

We consider reversible ergodic Markov chains with finite state space, and we introduce a new notion of quasi-stationary distribution that does not require the presence of any absorbing state. In our setting, the hitting time of the absorbing set is replaced by an optimal strong stationary time, representing the ``hitting time of the stationary distribution''. On the one hand, we show that our notion of quasi-stationary distribution corresponds to the natural generalization of the \emph{Yaglom limit}. On the other hand, similarly to the classical quasi-stationary distribution, we show that it can be written in terms of the eigenvectors of the underlying Markov kernel, and it is therefore amenable of a geometric interpretation. Moreover, we recover the usual exponential behavior that characterizes quasi-stationary distributions and metastable systems. We also provide some toy examples, which show that the phenomenology is richer compared to the absorbing case. Finally, we present some counterexamples, showing that the assumption on the reversibility cannot be weakened in general.