Large Deviation Principle for the Empirical Measures of Simple Random Walks on $\overline{\mathbb{Z}}$

Abstract

In this article we establish a large deviation principle for the empirical measures of a simple spatially inhomogeneous random walk on $\overline{\mathbb{Z}}$, the two-point compactification of $\mathbb{Z}$. The classical Donsker--Varadhan framework does not apply, since the random-walk kernel and the topology of $\overline{\mathbb{Z}}$ fall outside its standard assumptions. In certain regimes, the resulting rate function is non-convex on its effective domain. We also derive a large deviation principle for empirical means of observables $f:\mathbb{Z} \to \mathbb{R}^d$ admitting limits at $\pm\infty$. This result is optimal in the sense that in general, no large deviation principle holds for the larger class of bounded continuous functions on $\mathbb{Z}$.