Random walks with drift in the positive quadrant

TL;DR

This paper studies two-dimensional random walks with positive drift in the first quadrant, constructing harmonic functions, deriving tail asymptotics for exit times, and applying results to specific lattice walks.

Contribution

It introduces new methods for constructing harmonic functions and deriving asymptotics for walks with drift in the positive quadrant, including applications to lattice walks.

Findings

Constructed positive harmonic functions for walks with drift.

Derived tail asymptotics for exit times from the quadrant.

Applied results to specific lattice walks with step set {(1,-1),(1,1),(-1,1)}.

Abstract

We consider two dimensional random walks conditioned to stay in the positive quadrant. Assuming that the increments of the walk have finite second moments and that the drift vector is co-oriented with one of two axes, we construct positive harmonic functions for such walks and find tail asymptotics for the exit time from the positive quadrant. Moreover, we prove integral and local limit theorems. Finally, we apply our local limit theorems to singular lattice walks with steps $\{(1,-1),(1,1),(-1,1)\}$ and determine asymptotics for the number of walks of length $n$ which end on the line $\{(k,1),\ k\ge1\}$.