Limit Theorems in Hidden Markov Models

TL;DR

This paper establishes fundamental limit theorems such as the law of large numbers and central limit theorem for finite-state hidden Markov models, with applications to estimator convergence rates.

Contribution

It derives several limit theorems under mild conditions for hidden Markov models and applies them to analyze the convergence rate of maximum likelihood estimators.

Findings

Proves law of large numbers and CLT with error estimates for HMMs

Establishes an almost sure invariance principle and Chernoff bound variants

Determines the convergence rate of maximum likelihood estimators in HMMs

Abstract

In this paper, under mild assumptions, we derive a law of large numbers, a central limit theorem with an error estimate, an almost sure invariance principle and a variant of Chernoff bound in finite-state hidden Markov models. These limit theorems are of interest in certain ares in statistics and information theory. Particularly, we apply the limit theorems to derive the rate of convergence of the maximum likelihood estimator in finite-state hidden Markov models.