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Journal of the London Mathematical Society Advance Access published online on July 8, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn032
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© 2008 London Mathematical Society

Stability of the discrete time filter in terms of the tails of noise distributions

D. Crisan

Department of Mathematics
Huxley Building
Imperial College London
180 Queens Gate
London
SW7 2BZ
United Kingdom

K. Heine

Department of Mathematics
Tampere University of Technology
Korkeakoulunkatu 1
PO Box 553
FI-33101 Tampere
Finland
kari.heine@iki.fi

In recent years, the stability of discrete time filters has been a field of active research. By stability we mean that the effect of the possibly erroneous initial distribution in the filter eventually vanishes as time increases. One of the motivations for our interest in the stability is its close relation to the convergence of various numerical filter approximation schemes, for example, particle filters. In this paper, the main result states easily verifiable conditions that are sufficient for filter stability. Essentially, the conditions state that the filter is stable, if the observation noise is sufficiently light tailed compared with the randomness in the signal process. Compactness of the state space or ergodicity of the signal is not required.

2000 Mathematics Subject Classification 93E11, 93E15, 60G35, 62M20.

Received September 4, 2006; revised February 26, 2008;
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