Maximum likelihood estimator and its consistency for an $(L,1)$ random walk in a parametric random environment

TL;DR

This paper derives a maximum likelihood estimator for the environment parameter of an $(L,1)$ random walk in a random environment, demonstrating its explicit form and consistency through multitype branching process analysis.

Contribution

It introduces a novel MLE for the environment parameter based on a multitype branching process framework, extending previous work in the field.

Findings

Explicit computation of the invariant distribution of the multitype BPIRE.

Proof of the consistency of the maximum likelihood estimator.

Generalization of prior results by Comets et al.

Abstract

Consider an $(L,1)$ random walk in an i.i.d. random environment, whose environment involves certain parameter. We get the maximum likelihood estimator(MLE) of the environment parameter which can be written as functionals of a multitype branching process with immigration in a random environment(BPIRE). Because the offspring distributions of the involved multitype BPIRE are of the linear fractional type, the limit invariant distribution of the multitype BPIRE can be computed explicitly. As a result, we get the consistency of the MLE. Our result is a generalization of Comets et al. [Stochastic Process. Appl. 2014, 124, 268-288].