Necessary conditions for the invariant measure of a random walk to be a sum of geometric terms

TL;DR

This paper establishes necessary conditions for the invariant measure of a homogeneous random walk in the quarter-plane to be expressed as an infinite sum of geometric terms, highlighting structural and transition restrictions.

Contribution

It introduces new necessary conditions, including the requirement that geometric terms satisfy interior balance equations and restrictions on transition directions.

Findings

Geometric terms must satisfy interior balance equations.

Invariant measure geometric terms form finitely many coupled sets.

Transitions to north, northeast, east are not allowed.

Abstract

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as an infinite sum of geometric terms. We present necessary conditions for the invariant measure to be a sum of geometric terms. We demonstrate that, under a mild regularity condition, each geometric term must individually satisfy the balance equations in the interior of the state space. We show that the geometric terms in an invariant measure must be the union of finitely many pairwise-coupled sets of infinite cardinality. We further show that for the invariant measure to be a sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for an infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.