We obtain a perfect sampling characterization of weak ergodicity\udfor backward products of finite stochastic matrices, and equivalently,\udsimultaneous tail triviality of the corresponding nonhomogeneous\udMarkov chains. Applying these ideas to hidden Markov models, we\udshow how to sample exactly from the finite-dimensional conditional\uddistributions of the signal process given infinitely many observations,\udusing an algorithm which requires only an almost surely finite number\udof observations to actually be accessed. A notion of “successful”\udcoupling is introduced and its occurrence is characterized in terms\udof conditional ergodicity properties of the hidden Markov model and\udrelated to the stability of nonlinear filters.
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