Martin boundary of a killed random walk on a quadrant

Abstract

A complete representation of the Martin boundary of killed random walks on the quadrant ℕ^*×ℕ^* is obtained. It is proved that the corresponding full Martin compactification of the quadrant ℕ^*×ℕ^* is homeomorphic to the closure of the set {w=z/(1+|z|) : z∈ℕ^*×ℕ^*} in ℝ². The method is based on a ratio limit theorem for local processes and large deviation techniques.