Non parametric estimation for random walks in random environment

• Published in: Stochastic processes and their applications Vol. 128; no. 1; pp. 132 - 155
• Main Authors: Diel, Roland, Lerasle, Matthieu
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2018; Elsevier
• Subjects: Adaptive estimation; Mathematics; Non-parametric estimation; Oracle inequalities; Random walk in random environment; Statistics; secondary; Adaptive estimation; Random walk in random environment; Oracle inequalities; Non-parametric estimation; primary; and phrases : random walk in random environment; oracle inequalities; 60K37; adaptive estimation AMS 2010 subject classification : Primary 62G05; non-parametric estimation; Secondary 62E17
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2017.04.011

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.