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Invariant measures of the pair: State, approximation filtering process. (English) Zbl 0795.60028
The paper contains a proof and discussion of a theorem related to a problem in discrete time filtering which may be summarized as follows. The state is a Markov process with an initial law $$\nu$$. The filter is then also Markov. It is known that, if the state has a unique invariant measure, then, under certain conditions, the filter has in turn a unique invariant measure. Since $$\nu$$ is usually unknown, the initial law is approximated with the consequence that the filter based on this approximation is no longer Markov. It is shown that the couple formed by the state process and the filter with the approximated initial law is a homogeneous Markov process which, if the state process is aperiodic and if there are no transient states, has a unique invariant measure.

MSC:
 60G35 Signal detection and filtering (aspects of stochastic processes) 62M20 Inference from stochastic processes and prediction
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