Density estimation for RWRE

TL;DR

This paper develops a non-parametric density estimator for the environment in a random walk in random environment (RWRE), using beta-moments and the Goldenshluger-Lepski method, with theoretical bounds and simulation validation.

Contribution

It introduces a novel density estimation method for RWRE environments using beta-moments and adaptive selection, providing non-asymptotic bounds and empirical validation.

Findings

The estimator achieves accurate density recovery in RWRE.

The Goldenshluger-Lepski method effectively selects the beta-moment.

Simulation results confirm theoretical bounds.

Abstract

We consider the problem of non-parametric density estimation of a random environment from the observation of a single trajectory of a random walk in this environment. We first construct a density estimator using the beta-moments. We then show that the Goldenshluger-Lepski method can be used to select the beta-moment. We prove non-asymptotic bounds for the supremum norm of these estimators for both the recurrent and the transient to the right cases. A simulation study supports our theoretical findings.