Alternative constructions of a harmonic function for a random walk in a cone

TL;DR

This paper introduces two new methods for constructing positive harmonic functions for random walks in cones, removing previous restrictions and extending the applicability of earlier limit results to broader cone types.

Contribution

It provides novel constructions of harmonic functions that eliminate the need for strong extendability assumptions, broadening the class of cones where limit results hold.

Findings

Harmonic functions constructed without extendability assumptions

Results now apply to convex or star-like $C^2$ cones

Limit theorems remain valid under new constructions

Abstract

For a random walk killed at leaving a cone we suggest two new constructions of a positive harmonic function. These constructions allow one to remove a quite strong extendability assumption, which has been imposed in our previous paper (Denisov and Wachtel, 2015, Random walks in cones). As a consequence, all the limit results from that paper remain true for cones which are either convex or star-like and $C^2$.