Yaglom Limit via Holley Inequality

TL;DR

This paper uses Holley inequality to establish conditions for stochastic domination and uniqueness of the Yaglom limit in Markov chains with absorption, providing new proofs for classical results.

Contribution

It introduces a novel application of Holley inequality to characterize the Yaglom limit and minimal quasi-stationary distributions in partially ordered state spaces.

Findings

Stochastic domination of trajectories starting from minimal states

Uniqueness of the minimal quasi-stationary distribution

New proofs of classical results in the theory of quasi-stationary distributions

Abstract

Let $S$ be a countable set provided with a partial order and a minimal element. Consider a Markov chain on $S\cup\{0\}$ absorbed at $0$ with a quasi-stationary distribution. We use Holley inequality to obtain sufficient conditions under which the following hold. The trajectory of the chain starting from the minimal state is stochastically dominated by the trajectory of the chain starting from any probability on $S$, when both are conditioned to nonabsorption until a certain time. Moreover, the Yaglom limit corresponding to this deterministic initial condition is the unique minimal quasi-stationary distribution in the sense of stochastic order. As an application, we provide new proofs to classical results in the field.