Computing the Yaglom limit of Markov chains with a single exit state using their excursion measure

TL;DR

This paper establishes the existence of the Yaglom limit for certain Markov chains with a single exit state by analyzing their excursion measures, linking large deviations and quasi-stationary distributions.

Contribution

It introduces a novel approach using excursion measures to compute the Yaglom limit and minimal quasi-stationary distribution for Markov chains with a single exit state.

Findings

Proves the existence of the Yaglom limit in this setting.

Links the Yaglom limit to large deviations of the inverse local time.

Provides a method to compute the minimal quasi-stationary distribution.

Abstract

We prove in this article the existence of the Yaglom limit for Markov chains on discrete state spaces in the setting where the absorbing state is accessible from a single non-absorbing state. We use a representation of the trajectories of this process by its excursion away from death, that allows us to link the Yaglom limit with the large deviations behaviour of the inverse of its local time at the exit state, and to compute its minimal quasi-stationary distribution with its excursion measure.