Non parametric estimation for random walks in random environment

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.