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

TL;DR

This paper investigates the asymptotic behavior of the maximum likelihood estimator in hidden Markov models indexed by binary trees with general metric space states, establishing strong consistency and asymptotic normality.

Contribution

It extends the theoretical understanding of MLE properties in complex tree-indexed HMMs with general state spaces, under standard assumptions.

Findings

Proves strong consistency of MLE in tree-indexed HMMs.

Establishes asymptotic normality of the MLE.

Works under both stationary and non-stationary regimes.

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 properties of the initial distribution conditional on the observations. The proofs rely on ergodic theorems for Markov chain indexed by trees with neighborhood-dependent functions.