Maximum likelihood estimator consistency for recurrent random walk in a parametric random environment with finite support

TL;DR

This paper develops a maximum likelihood estimation method for one-dimensional recurrent random walks in i.i.d. random environments with finite support, addressing challenges in convergence and asymptotics.

Contribution

It introduces a novel MLE approach for RWRE with finite support, accounting for nonstandard limit behavior and second order asymptotics.

Findings

The MLE converges under specific conditions despite nondegenerate limits.

Second order asymptotics are essential for parameter identification.

Numerical experiments demonstrate the method's effectiveness.

Abstract

We consider a one-dimensional recurrent random walk in random environment (RWRE) when the environment is i.i.d. with a parametric, finitely supported distribution. Based on a single observation of the path, we provide a maximum likelihood estimation procedure of the parameters of the environment. Unlike most of the classical maximum likelihood approach, the limit of the criterion function is in general a nondegenerate random variable and convergence does not hold in probability. Not only the leading term but also the second order asymptotics is needed to fully identify the unknown parameter. We present different frameworks to illustrate these facts. We also explore the numerical performance of our estimation procedure.