Consistency of the maximum likelihood estimator for general hidden Markov models

Randal Douc, Eric Moulines, Jimmy Olsson, Ramon van Handel

Research output: Contribution to journalArticlepeer-review


Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete separable metric space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state space models, as well as general results on linear Gaussian state space models and finite state models. A novel aspect of our approach is an information-theoretic technique for proving identifiability, which does not require an explicit representation for the relative entropy rate. Our method of proof could therefore form a foundation for the investigation of MLE consistency in more general dependent and non-Markovian time series. Also of independent interest is a general concentration inequality for V-uniformly ergodic Markov chains.
Original languageEnglish
Pages (from-to)474-513
JournalAnnals of Statistics
Issue number1
Publication statusPublished - 2011

Subject classification (UKÄ)

  • Probability Theory and Statistics


  • Hidden Markov models
  • maximum likelihood estimation
  • strong
  • consistency
  • V-uniform ergodicity
  • concentration inequalities
  • state
  • space models


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