Non parametric estimation for random walks in random environment

We consider a random walk in i.i.d. random environment with distribution ν on Z. The problem we are interested in is to provide an estimator of the cumulative distribution function (c.d.f.) F of ν from the observation of one trajectory of the random walk. For that purpose we first estimate the moments of ν, then combine these moment estimators to obtain a collection of estimators (F M n) M ≥1 of F , our final estimator is chosen among this collection by Lepskii's method. This estimator is therefore easily computable in practice. 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 optimal when the chain realizes a trade-off between a fast exploration of the sites, allowing to get more informations and a larger number of visits of each sites, allowing a better recovery of the environment itself.