Asymptotically stable random walks of index $1<\alpha <2$ killed on a finite set

Abstract

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.

51 pages: this is an improved version of arXiv:1808.01484, where the RW is supposed to belong to the domain of normal attraction to a stable law, while in the present version the restriction 'normal' is removed