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Convergence of filters with applications to the Kalman-Bucy case

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1 Author(s)
Goggin, E.M. ; Dept. of Math., Iowa State Univ., Ames, IA, USA

For each N, and each fixed time T, a signal XN and a `noisy' observation YN are defined by a pair of stochastic difference equations. Under certain conditions (XN, YN) converges in distribution to (X, Y, where dX(t)= f(t, X(t))dt+dV( t), dY(t)=g(t, X( t))dt+dW(t). Conditions are found under which convergence in distribution of the conditional expectations E{F(XN)|YN} to E{F(X)|Y} follows, for every bounded continuous function F. The case in which the conditional expectations still converge but the limit is not E{ F(X)|Y} is also studied. In the situation where f and g are linear functions of X, an examination of this limit leads to a Kalman-Bucy-type estimate of X N which is asymptotically optimal and is an improvement on the usual Kalman-Bucy estimate

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Information Theory, IEEE Transactions on  (Volume:38 ,  Issue: 3 )