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Discrete time nonlinear filters with informative observations are stable


 
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1. Title Title of document Discrete time nonlinear filters with informative observations are stable
 
2. Creator Author's name, affiliation, country Ramon Van Handel; Princeton University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) nonlinear filtering; prediction; asymptotic stability; hidden Markov models
 
3. Subject Subject classification 60J05; 62M20; 93E11; 93E15
 
4. Description Abstract The nonlinear filter associated with the discrete time signal-observation model $(X_k,Y_k)$ is known to forget its initial condition as $k\to\infty$ regardless of the observation structure when the signal possesses sufficiently strong ergodic properties. Conversely, it stands to reason that if the observations are sufficiently informative, then the nonlinear filter should forget its initial condition regardless of any properties of the signal. We show that for observations of additive type $Y_k=h(X_k)+\xi_k$ with invertible observation function $h$ (under mild regularity assumptions on $h$ and on the distribution of the noise $\xi_k$), the filter is indeed stable in a weak sense without any assumptions at all on the signal process. If the signal satisfies a uniform continuity assumption, weak stability can be strengthened to stability in total variation.
 
5. Publisher Organizing agency, location
 
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7. Date (YYYY-MM-DD) 2008-11-14
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1423
 
10. Identifier Digital Object Identifier 10.1214/ECP.v13-1423
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 13
 
12. Language English=en
 
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