We give simple conditions that ensure exponential forgetting of the initial conditions of the filter for general state-space hidden Markov chain. The proofs are based on the coupling argument applied to the posterior Markov kernels. These results are useful both for filtering hidden Markov models using approximation methods (e.g., particle filters) and for proving asymptotic properties of estimators. The results are general enough to cover models like the Gaussian state space model, without using the special structure that permits the application of the Kalman filter.
"Forgetting of the initial condition for the filter in general state-space hidden Markov chain: a coupling approach." Electron. J. Probab. 14 27 - 49, 2009. https://doi.org/10.1214/EJP.v14-593