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We give simple conditions that ensure exponential forgetting of the initialconditions of the filter for general state-space hidden Markov chain. Theproofs are based on the coupling argument applied to the posterior Markovkernels. These results are useful both for filtering hidden Markov models usingapproximation methods (e.g., particle filters) and for proving asymptoticproperties of estimators. The results are general enough to cover models likethe Gaussian state space model, without using the special structure thatpermits the application of the Kalman filter.

Forgetting of the initial condition for the filter in general state-space hidden Markov chain: a coupling approach