# Non parametric estimation for random walks in random environment

*Roland Diel* and
*Matthieu Lerasle*

*Stochastic Processes and their Applications*, 2018, vol. 128, issue 1, 132-155

**Abstract:**
We investigate the problem of estimating the cumulative distribution function (c.d.f.) F of a distribution ν from the observation of one trajectory of the random walk in i.i.d. random environment with distribution ν on Z. We first estimate the moments of ν, then combine these moment estimators to obtain a collection of estimators (F̂nM)M≥1 of F, our final estimator is chosen among this collection by Goldenshluger–Lepski’s method. This estimator is easily computable. We derive convergence rates for this estimator depending on the Hölder regularity of F and on the divergence rate of the walk. Our rate is minimal when the chain realizes a trade-off between a fast exploration of the sites, allowing to get more information and a larger number of visits of each site, allowing a better recovery of the environment itself.

**Keywords:** Random walk in random environment; Non-parametric estimation; Oracle inequalities; Adaptive estimation (search for similar items in EconPapers)

**Date:** 2018

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:spapps:v:128:y:2018:i:1:p:132-155

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