One dimensional random walks killed on a finite set

We study the transition probability, say pAn(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 σ2. We show that pAn(x,y) behaves like [gA+(x)ĝA+(y)+gA−(x)ĝA−(y)](σ2/2n)pn(y−x) uniformly in the regime characterized by the conditions ∣x∣∨∣y∣=O(n) and ∣x∣∧∣y∣=o(n) generally if xy>0 and under a mild additional assumption about the walk if xy<0. Here pn(y−x) is the transition kernel of the random walk (without killing); gA± are the Green functions for the 'exterior' of A with 'pole at ±∞' normalized so that gA±(x)∼2∣x∣/σ2 as x→±∞; and ĝA± are the corresponding Green functions for the time-reversed walk.