Discrete time nonlinear filters with informative observations are stable

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Abstract

The nonlinear filter associated with the discrete time signal-observation model (Xk, Yk) 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 Yk = h(Xk) + ξk with invertible observation function h (under mild regularity assumptions on hand on the distribution of the noise ξk), 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 languageEnglish (US)
Pages (from-to)562-575
Number of pages14
JournalElectronic Communications in Probability
Volume13
DOIs
StatePublished - 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

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