Yaglom limit via Holley inequality

Abstract

Let SS be a countable set provided with a partial order and a minimal element. Consider a Markov chain on S∪{0}S∪{0} absorbed at 00 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 SS, 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.