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

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. For that purpose, we adapt the method developed in the ballistic case by Comets et al. (2014) and Falconnet et al. (2014). Using a supplementary assumption due to the special nature of the sub-ballistic regime, we prove consistency and asymptotic normality as the distant site tends to infinity. To emphasize 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 the ballistic regime, the Fisher information is infinite. We also explore the numerical performance of our estimation procedure. © 2014, Allerton Press, Inc.