Passage-times for partially-homogeneous reflected random walks on the quadrant

TL;DR

This paper classifies recurrence and transience of partially-homogeneous reflected random walks in the quadrant, providing power-law bounds on passage times, especially in the zero-drift interior case, with applications to multidimensional Lindley processes.

Contribution

It offers a novel classification of recurrence and transience for these walks and derives quantitative passage-time bounds in the zero-drift interior setting.

Findings

Classification depends on interior covariance and boundary drifts.

Power-law bounds on passage-time tails are established.

Application to multidimensional Lindley processes with new quantitative results.

Abstract

We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by finitely-many transition laws near each boundary, together with an interior transition law that applies at sufficient distance from both boundaries. Under mild assumptions, in the (most subtle) setting in which the mean drift in the interior is zero, we classify recurrence and transience and provide power-law bounds on tails of passage times; the classification depends on the interior covariance matrix, the (finitely many) drifts near the boundaries, and stationary distributions derived from two one-dimensional Markov chains associated to each of the two boundaries. As an application, we consider reflected random walks related to multidimensional variants of the Lindley process, for which the recurrence question was studied recently by Peign\'e and Woess (Ann. Appl. Probab., vol. 31, 2021) using different methods, but for which no previous quantitative results on passage-times appear to be known.