Uniform convergence to the Q-process

Nicolas Champagnat (1, 2) , Denis Villemonais (1, 3, 2)

1 TOSCA - TO Simulate and CAlibrate stochastic models
2 IECL - Institut Élie Cartan de Lorraine
3 ENSMN - Ecole Nationale Supérieure des Mines de Nancy

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.

Mots clés: quasi-stationary distribution; Q-process; uniform exponential mixing property; conditional ergodic theorem

Domaines: Probabilités [math.PR]

Nicolas Champagnat, Denis Villemonais. Uniform convergence to the Q-process. Electronic Communications in Probability, 2017, 22 (33), pp.1-7. ⟨10.1214/17-ECP63⟩. ⟨hal-01395727v2⟩