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Yaglom limit via Holley inequality. (English) Zbl 1321.60142

Summary: 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 the Holley inequality to obtain sufficient conditions under which the following holds: 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.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60E15 Inequalities; stochastic orderings
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