The quenched limiting distributions of a one-dimensional random walk in random scenery

Nadine Guillotin-Plantard,Bruno Schapira,Yueyun Hu

doi:10.1214/ecp.v18-2918

Abstract

For a one-dimensional random walk in random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen's functional law of iterated logarithm. As a consequence, conditioned on the random scenery, the one dimensional RWRS does not converge in law, in contrast with the multi-dimensional case.

Highlights

Random walks in random sceneries were introduced independently by Kesten and Spitzer [9]and by Borodin [3, 4]
In [8], the case of the planar random walk was studied, the authors proved a quenched version of the annealed central limit theorem obtained by Bolthausen in [2]
We prove that under these assumptions, there is no quenched distributional limit theorem for K

Summary

Introduction

Random walks in random sceneries were introduced independently by Kesten and Spitzer [9]and by Borodin [3, 4]. In [8], the case of the planar random walk was studied, the authors proved a quenched version of the annealed central limit theorem obtained by Bolthausen in [2]. | ξ), the limit points of the law of Kn, as n → ∞, under the topology of weak convergence of measures, are equal to the set of the laws of random variables in ΘB, with

Results

Conclusion