Random walks in cones

TL;DR

This paper investigates the long-term behavior of multidimensional random walks confined within cones, establishing tail asymptotics, limit theorems, and constructing harmonic functions under minimal assumptions, with applications to ordered walks and lattice paths.

Contribution

It introduces a novel approach to analyze random walks in cones by constructing harmonic functions with minimal moment conditions and applying strong approximation techniques.

Findings

Derived tail asymptotics for exit times

Proved integral and local limit theorems for conditioned walks

Provided applications to ordered walks and lattice enumeration

Abstract

We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results to ordered random walks and lattice path enumeration.