A primer on perfect simulation

Perfect simulations or exact sampling are labels for a recently developed set of techniques designed to produce output whose distribution is guaranteed to follow a given probability law. These techniques are particularly useful in relation with Markov Chain Monte Carlo, and their range of applicability is rapidly growing (see Green and Murdoch (1999), Section 1.3, and Mira, Møller and Roberts (1999) and Møller and Nicholls (1999) and references therein, or visit the site http://dimacs.rutgers.edu/~dbwilson/exact). Several techniques have been suggested recently in the literature. The outbreak of these subject come with Propp and Wilson (1996) paper where they suggest a practical method of achieving a perfect sample of a Markov chain with finite state space. Their Coupling from the Past (CFTP) algorithm have been applied for infinite (or huge) state spaces require a monotonicity property : there must exist a “maximal” and a “minimal” states and a coupling such that the coalescence of trajectories starting from these two states imply the coalescence of all other trajectories (“monotone coupling”). Examples of processes with this property include Glauber dynamics of spin systems with the FKG property (Propp and Wilson, 1996) and attractive point processes (Kendall, 1997; Kendall, 1998). In fact, through a minor modification the algorithm is also applicable to repulsive point processes (Kendall, 1998; Häggström and Nelander, 1998). One of the biggest problem with CFTP technique is that it has the so called impatient-user bias. Fill (1998) introduced a technique, free of the