  This paper generalizes the work of Kendall [Electron. Comm. Probab. 9 (2004)
140--151], which showed that perfect simulation, in the form of dominated
coupling from the past, is always possible (although not necessarily practical)
for geometrically ergodic Markov chains. Here, we consider the more general
situation of positive recurrent chains and explore when it is possible to
produce such a simulation algorithm for these chains. We introduce a class of
chains which we name tame, for which we show that perfect simulation is
possible.

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