  For a random walk on the integer lattice $\mathbb{Z}$ that is attracted to a
strictly stable process with index $\alpha\in (1, 2)$ we obtain the asymptotic
form of the transition probability for the walk killed when it hits a finite
set. The asymptotic forms obtained are valid uniformly in a natural range of
the space and time variables. The situation is relatively simple when the limit
stable process has jumps in both positive and negative directions; in the other
case when the jumps are one sided rather interesting matters are involved and
detailed analyses are necessitated.
