Uniform convergence to the Q-process

TL;DR

This paper investigates the convergence rate of conditioned Markov processes to their Q-processes and establishes conditions for uniform convergence and exponential ergodicity, with applications to a conditional ergodic theorem.

Contribution

It provides new quantitative bounds on convergence speed and characterizes conditions for uniform convergence and ergodicity of conditioned processes.

Findings

Quantifies convergence speed to the Q-process under certain assumptions.

Proves uniform convergence implies existence and exponential convergence to a quasi-stationary distribution.

Provides a conditional ergodic theorem as an application.

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.