Asymptotic properties of the maximum likelihood estimator for Hidden Markov Models indexed by binary trees

Julien Weibel

Abstract:

We consider hidden Markov models indexed by a binary tree where the hidden state space
is a general metric space. We study the maximum likelihood estimator (MLE) of the model
parameters based only on the observed variables. In both stationary and non-stationary
regimes, we prove strong consistency and asymptotic normality of the MLE under standard
assumptions. Those standard assumptions imply uniform exponential memorylessness prop-
erties of the initial distribution conditional on the observations. The proofs rely on ergodic
theorems for Markov chain indexed by trees with neighborhood-dependent functions.