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

TL;DR

This paper investigates the limiting behavior of a one-dimensional random walk in random scenery, revealing that it does not converge in law when conditioned on the scenery, unlike higher-dimensional cases.

Contribution

It applies Strassen's functional law of the iterated logarithm to establish quenched weak limits for the one-dimensional RWRS, a novel approach in this context.

Findings

One-dimensional RWRS does not converge in law conditioned on scenery.

The method uses Strassen's functional law of the iterated logarithm.

Contrasts with multi-dimensional RWRS behavior.

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 the 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.