Martin boundary of a killed random walk on a quadrant
The authors derive a complete representation of the Martin boundary of killed random walks on the quadrant \(\mathbb{N}^{\ast }\times \mathbb{N}^{\ast }\). It is proved that the corresponding full Martin compactification of the quadrant \(\mathbb{N}^{\ast }\times \mathbb{N}^{\ast } \) is homeomorphic to the closure of the set \(\left\{ w=z/\left( 1+\left| z\right| \right) :z\in \mathbb{N}^{\ast }\times \mathbb{N} ^{\ast }\right\} \) in \( \mathbb{R} ^{2}\). The method is based on a ratio limit theorem for local processes and large deviation techniques.
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