Perfect simulation and backward coupling

Algorithms for perfect or exact simulation of random samples from the invariant measure of a Markov chain have received considerable recent attention following the introduction of the “coupling-from-the-past” (CFTP) technique of Propp and Wilson. Here we place such algorithms in the context of backward coupling of stochastically recursive sequences. We show that although general backward couplings can be constructed for chains with finite mean forward coupling times, and can even be thought of as extending the classical “Loynes schemes” from queueing theory, successful “vertical” CFTP algorithms such as those of Propp and Wilson can be constructed if and only if the chain is uniformly geometric ergodic. We also relate the convergence moments for backward coupling methods to those of forward coupling times: the former typically lose at most one moment compared to the latter