  We obtain a perfect sampling characterization of weak ergodicity for backward
products of finite stochastic matrices, and equivalently, simultaneous tail
triviality of the corresponding nonhomogeneous Markov chains. Applying these
ideas to hidden Markov models, we show how to sample exactly from the
finite-dimensional conditional distributions of the signal process given
infinitely many observations, using an algorithm which requires only an almost
surely finite number of observations to actually be accessed. A notion of
"successful" coupling is introduced and its occurrence is characterized in
terms of conditional ergodicity properties of the hidden Markov model and
related to the stability of nonlinear filters.
