We consider a one-dimensional, nearest-neighbour, recurrent random walk (Xₖ), k n, in a random i. i. d. environment (RWRE). Assuming that the environment has finite support, which is treated as a parameter, we establish the Local Asymptotic Mixed Normality (LAMN) property for this parameter. The convergence rate is n, and the asymptotic Fisher information is random and expressed in terms of the invariant measure in the infinite valley, as introduced in Gantert et al. , 2010. We further show that the Maximum Likelihood Estimator (MLE) of the support parameter converges to a mixture of normal distributions and is asymptotically efficient. We further show that the Maximum Likelihood Estimator (MLE) of the support parameter converges to a mixture of normal distributions and is asymptotically efficient. The proofs are based on a recent result by Comets et al. , 2024, which extends to recurrent RWRE the method of the ``environment viewed from the particle'', originally introduced in KozMol for the transient ballistic RWRE.
Loukianova et al. (Wed,) studied this question.