Martin boundary of a killed random walk on a half-space

Abstract

A complete representation of the Martin boundary of killed random walks on a half-space ${\mathbb{Z}}^{d-1}\times {\mathbb{N}}^{\ast }$ is obtained. In particular, it is proved that the corresponding Martin boundary is homemorphic to the half-sphere ${\mathcal{S}}_{+}^d=\{z\in {\mathbb{R}}^{d-1}\times {\mathbb{R}}_{+}:|z|=1\}$. The method is based on a combination of ratio limits theorems and large deviation techniques.

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