A central limit and Berry-Esseen theorem for continuous-time Markov processes conditioned not to be absorbed

This paper aims to establish a central limit and Berry-Esseen-like theorem for Markov processes conditioned not to be absorbed. First, we prove that a central limit theorem holds true for the $Q$-process, under general criteria on its exponential ergodicity. Then, we prove that the Kolmogorov distance between the conditional distribution of the renormalized centered empirical mean for the absorbed process and the one for the $Q$-process decays as $1/\sqrt{t}$