## 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 →∞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}) + ξ_{k} with invertible observation function h (under mild regularity assumptions on hand on the distribution of the noise ξ_{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.

Original language | English (US) |
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Pages (from-to) | 562-575 |

Number of pages | 14 |

Journal | Electronic Communications in Probability |

Volume | 13 |

DOIs | |

State | Published - Jan 1 2008 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- Asymptotic stability
- Hidden Markov models
- Nonlinear filtering
- Prediction