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Uniform convergence to the Q-process

Nicolas Champagnat 1, 2 Denis Villemonais 1, 3, 2
1 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : The first aim of the present note is to quantify the speed of convergence of a conditioned process toward its Q-process under suitable assumptions on the quasi-stationary distribution of the process. Conversely, we prove that, if a conditioned process converges uniformly to a conservative Markov process which is itself ergodic, then it admits a unique quasi-stationary distribution and converges toward it exponentially fast, uniformly in its initial distribution. As an application, we provide a conditional ergodic theorem.
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Nicolas Champagnat, Denis Villemonais. Uniform convergence to the Q-process. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2017, 22 (33), pp.1-7. ⟨10.1214/17-ECP63⟩. ⟨hal-01395727v2⟩

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