One dimensional random walk killed on a finite set

We study the transition probability, say $p_A^n(x,y)$, of a one-dimensional random walk on the integer lattice killed when entering into a non-empty finite set $A$. The random walk is assumed to be irreducible and have zero mean and a finite variance ${\sigma }^2$. We derive the asymptotic form of $p_A^n(x,y)$ for large $n$ valid uniformly in the regime characterized by the conditions $\mathrm{\mid }x\mathrm{\mid }\vee \mathrm{\mid }y\mathrm{\mid }=O(\sqrt{n})$ and $\mathrm{\mid }x\mathrm{\mid }\wedge \mathrm{\mid }y\mathrm{\mid }=o(\sqrt{n})$, in which $p^A_t({\bf x},{\bf y})$behavesforlarge$n$like$[g_A^{+}(x)\hat g_{A}^{,+}(y) + g_A^-(x)\hat g_{A}^{,-}(y)] (\sigma^{2}/2n) p^n(y-x)$\mathrm{.}Here$p^n(y-x)$ is the transition kernel of the random walk (without killing); $g_A^{\pm }$ are the Green functions for the “exterior” of $A$ with “pole at $\pm \mathrm{\infty }$” normalized so that $g_A^{\pm }(x)\sim 2\mathrm{\mid }x\mathrm{\mid }\mathrm{/}{\sigma }^2$ as $x\to \pm \mathrm{\infty }$; and $\hat g_A^{, \pm}$ are the corresponding Green functions for the time-reversed walk.