LAMN property for recurrent Random Walk in Random Environment
Oleg Loukianov, Dasha Loukianova, Thi Phuong Thuy Vo · HAL (Le Centre pour la Communication Scientifique Directe) · 2025
We consider a one-dimensional, nearest-neighbour, recurrent random walk in a random environment (RWRE). Assuming that the environment has a finite support, which is treated as a parameter, we establish the Local Asymptotic Mixed Normality (LAMN) property for this parameter. The asymptotic random Fisher information is 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. The proofs rely 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 Kozlov and Molchanov (1984) for the transient ballistic RWRE.