We consider the problem of nonparametric density estimation of a random environment from the observation of a single trajectory of a random walk in this environment. We build several density estimators using the beta-moments of this distribution. Then we apply the Goldenschluger-Lepski method to select an estimator satisfying an oracle type inequality. We obtain non-asymptotic bounds for the supremum norm of these estimators that hold when the RWRE is recurrent or transient to the right. A simulation study supports our theoretical findings.

中文翻译：

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RWRE的密度估计

我们通过在这种环境中观察随机行走的单个轨迹来考虑随机环境的非参数密度估计问题。我们使用此分布的β矩建立了几个密度估计量。然后，我们应用Goldenschluger-Lepski方法来选择满足oracle类型不等式的估计量。我们获得了这些估计量的最高范数的非渐近边界，当RWRE反复出现或向右瞬变时，这些估计值成立。模拟研究支持了我们的理论发现。