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

## TL;DR

This paper derives the asymptotic transition probabilities for a class of stable-like random walks on integers that are terminated upon hitting a finite set, with detailed analysis depending on jump directions.

## Contribution

It provides the first detailed asymptotic analysis of transition probabilities for stable-like random walks killed at finite sets, covering both two-sided and one-sided jump cases.

## Key findings

- Asymptotic transition probabilities are obtained uniformly in space and time.
- Different behaviors are characterized for two-sided and one-sided jump stable processes.
- Results extend understanding of killed stable processes on integer lattices.

## 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.