Random walks in cones revisited

TL;DR

This paper introduces a new method for analyzing multidimensional random walks in cones, enabling the construction of harmonic functions under minimal conditions and providing refined behavior insights near boundaries.

Contribution

The paper presents a novel approach to construct harmonic functions in Lipschitz cones with minimal moment assumptions, improving understanding of boundary behavior and limit theorems.

Findings

Constructed positive harmonic functions in Lipschitz cones.

Achieved more precise boundary behavior analysis.

Proved new limit theorems under relaxed moment conditions.

Abstract

In this paper we continue our study of a multidimensional random walk with zero mean and finite variance killed on leaving a cone. We suggest a new approach that allows one to construct a positive harmonic function in Lipschitz cones under minimal moment conditions. This approach allows one also to obtain more accurate information about the behaviour of the harmonic function not far from the boundary of the cone. We also prove limit theorems under new moment conditions.