Green function for an asymptotically stable random walk in a half space

We consider an asymptotically stable multidimensional random walk $S(n)=(S_1(n),\dots ,S_d(n))$. Let ${\tau }_x:=\min \{n>0:x_1+S_1(n)\le 0\}$ be the first time the random walk $S(n)$ leaves the upper half-space. We obtain the asymptotics of $p_n(x,y):=P(x+S(n)\in y+\mathrm{\Delta },{\tau }_x>n)$ as $n$ tends to infinity, where $\mathrm{\Delta }$ is a fixed cube. From that we obtain the local asymptotics for the Green function $G(x,y):={\sum }_n{p}_n(x,y)$, as $\mathrm{\mid }y\mathrm{\mid }$ and/or $\mathrm{\mid }x\mathrm{\mid }$ tend to infinity.