This note extends the work of Foss and Tweedie (1998), who showed that availability of the classic Coupling from the Past (CFTP) algorithm of Propp and Wilson (1996) is essentially equivalent to uniform ergodicity for a Markov chain (see also Hobert and Robert, 2004). In this note we show that all geometrically ergodic chains possess dominated CFTP algorithms (not necessarily practical!) which are rather closely connected to Foster-Lyapunov criteria. Hence geometric ergodicity implies dominated CFTP.
"Geometric Ergodicity and Perfect Simulation." Electron. Commun. Probab. 9 140 - 151, 2004. https://doi.org/10.1214/ECP.v9-1117