Stable random walks in cones

TL;DR

This paper studies the behavior of multidimensional stable-like random walks confined in cones, constructing harmonic functions, analyzing exit time tails, and establishing limit theorems for their exit behavior.

Contribution

It introduces a method to construct harmonic functions for stable-like random walks in cones and derives asymptotic tail estimates and limit theorems for exit times.

Findings

Derived the asymptotic tail distribution of exit times.

Constructed positive harmonic functions for the random walk.

Proved a conditional functional limit theorem for exit behavior.

Abstract

In this paper we consider a multidimensional random walk killed on leaving a right circular cone with a distribution of increments belonging to the normal domain of attraction of an $\alpha$-stable and rotationally-invariant law with $\alpha \in (0,2)\setminus \{1\}$. Based on Bogdan et al. (2018) describing the tail behaviour of the exit time of $\alpha$-stable process from a cone and using some properties of Martin kernel of the isotropic $\alpha$-stable process, in this paper we construct a positive harmonic function of the discrete time random walk under consideration. Then we find the asymptotic tail of the distribution of the exit time of this random walk from the cone. We also prove the corresponding conditional functional limit theorem.