Sinai's Walk : a Statistical Aspect (preliminary Version)

In this paper we are interested in Sinai’s walk i.e a one dimensional random walk in random environment with three conditions on the random environment: two necessaries hypothesis to get a recurrent process (see [Solomon(1975)]) which is not a simple random walk and an hypothesis of regularity which allows us to have a good control on the fluctuations of the random environment. The asymptotic behavior of such walk has been understood by [Sinai(1982)] : this walk is subdiffusive and at an instant n it is localized in the neighborhood of a well defined point of the lattice. It is well known, see (Zeitouni [2001] for a survey) that this behavior is strongly dependent of the random environment or, equivalently, by the associated random potential defined Section 1.2. The question we solve here is the following: given a single trajectory of a random walk (Xk, 1 ≤ k ≤ n) where n is fixed, can we estimate the trajectory of the random potential where the walk lives ? Let us remark that the law of this potential is unknown as-well. In their paper, [Adelman and Enriquez(2004)] answer this question in the parametrical case when the law of the random environment is defined by two parameters. On the other hand, our approach is a non-parametrical approach based on good properties of the local time of the random walk. We are able to reconstruct the random potential in a significant interval where the walk spends most of its time. The key point of this paper is that if we impose to the local time to be large enough but negligible comparing to the maximum of the local time then this will directly implies conditions on the random potential. Notice that, this aspect of looking the random