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## Consistency of the Maximum Likelihood Estimator for general hidden Markov models

Randal Douc
• Function : Author
• PersonId : 859568
Eric Moulines
• Function : Author
• PersonId : 849229
Jimmy Olsson
• Function : Author
• PersonId : 835141
Ramon van Handel
• Function : Author
• PersonId : 865692

#### Abstract

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.

### Dates and versions

hal-00442774 , version 1 (22-12-2009)

### Identifiers

• HAL Id : hal-00442774 , version 1
• ARXIV :

### Cite

Randal Douc, Eric Moulines, Jimmy Olsson, Ramon van Handel. Consistency of the Maximum Likelihood Estimator for general hidden Markov models. 2009. ⟨hal-00442774⟩

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