Geometric Ergodicity and Perfect Simulation
  This note extends the work of Foss and Tweedie (1997), who showed that availability of the classic Coupling from The Past algorithm of Propp and Wilson (1996) is essentially equivalent to uniform ergodicity for a Markov chain (see also HobertRobert, 2004). In this note we show that all geometrically ergodic chains possess dominated Coupling from The Past algorithms (not necessarily practical!) which are rather closely connected to Foster-Lyapunov criteria. 
Wilfrid S. Kendall