Parametric estimation of a one-dimensional ballistic random walk in a Markov environment

TL;DR

This paper develops a method for estimating the parameters of a Markov environment from a single observed trajectory of a ballistic random walk, proving statistical properties like consistency and efficiency.

Contribution

It reformulates the problem as a hidden Markov model and proves positive Harris recurrence of the underlying chain, extending estimation techniques to this setting.

Findings

Maximum likelihood estimator is consistent, asymptotically normal, and efficient.

The underlying bivariate Markov chain is positive Harris recurrent.

Results apply to models like DNA unzipping.

Abstract

We focus on the parametric estimation of the distribution of a Markov environment from the observation of a single trajectory of a one-dimensional nearest-neighbor path evolving in this random environment. In the ballistic case, as the length of the path increases, we prove consistency, asymptotic normality and efficiency of the maximum likelihood estimator. Our contribution is two-fold: we cast the problem into the one of parameter estimation in a hidden Markov model (HMM) and establish that the bivariate Markov chain underlying this HMM is positive Harris recurrent. We provide different examples of setups in which our results apply, in particular that of DNA unzipping model, and we give a simple synthetic experiment to illustrate those results.