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Geometric ergodicity and perfect simulation. (English) Zbl 1061.60070
Summary: This note extends the work of S. G. Foss and R. L. Tweedie [Commun. Stat., Stochastic Models 14, No. 1/2, 187–203 (1998; Zbl 0934.60088)], who showed that availability of the classic coupling from the past (CFTP) algorithm of J. G. Propp and D. B. Wilson [Random Struct. Algorithms 9, No. 1/2, 223–252 (1996; Zbl 0859.60067)] is essentially equivalent to uniform ergodicity for a Markov chain [see also J. P. Hobert and C. P. Robert, Ann. Appl. Probab. 14, No. 3, 1295–1305 (2004; Zbl 1046.60062)]. 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.

60J05 Discrete-time Markov processes on general state spaces
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