Maximum likelihood estimation in the context of a sub-ballistic random walk in a parametric random environment

TL;DR

This paper investigates the maximum likelihood estimator for a parameter in a one-dimensional sub-ballistic random walk within a parametric i.i.d. environment, establishing its consistency and asymptotic normality under specific conditions.

Contribution

It extends existing methods to the sub-ballistic regime, proving the MLE's properties and analyzing its behavior in the Temkin model with unknown support.

Findings

MLE is consistent in the sub-ballistic regime

Asymptotic normality of the MLE is established

Fisher information is infinite in the Temkin model with unknown support

Abstract

We consider a one dimensional sub-ballistic random walk evolving in a parametric i.i.d. random environment. We study the asymptotic properties of the maximum likelihood estimator (MLE) of the parameter based on a single observation of the path till the time it reaches a distant site. In that purpose, we adapt the method developed in the ballistic case by Comets et al (2014) and Falconnet, Loukianova and Matias (2014). Using a supplementary assumption due to the specificity of the sub-ballistic regime, we prove consistency and asymptotic normality as the distant site tends to infinity. To emphazis the role of the additional assumption, we investigate the Temkin model with unknown support, and it turns out that the MLE is consistent but, unlike in the ballistic regime, the Fisher information is infinite. We also explore the numerical performance of our estimation procedure.