Uniform observability of hidden Markov models and filter stability for unstable signals

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Abstract

A hidden Markov model is called observable if distinct initial laws give rise to distinct laws of the observation process. Observability implies stability of the nonlinear filter when the signal process is tight, but this need not be the case when the signal process is unstable. This paper introduces a stronger notion of uniform observability which guarantees stability of the nonlinear filter in the absence of stability assumptions on the signal. By developing certain uniform approximation properties of convolution operators, we subsequently demonstrate that the uniform observability condition is satisfied for various classes of filtering models with white-noise type observations. This includes the case of observable linear Gaussian filtering models, so that standard results on stability of the Kalman-Bucy filter are obtained as a special case.

Original languageEnglish (US)
Pages (from-to)1172-1199
Number of pages28
JournalAnnals of Applied Probability
Volume19
Issue number3
DOIs
StatePublished - Jun 2009

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Asymptotic stability
  • Hidden markov models
  • Merging of probability measures
  • Nonlinear filtering
  • Observability
  • Prediction
  • Uniform approximation

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