  We consider the invariant measure of homogeneous random walks in the
quarter-plane. In particular, we consider measures that can be expressed as a
finite linear combination of geometric terms and present conditions on the
structure of these linear combinations such that the resulting measure may
yield an invariant measure of a random walk. We demonstrate that each geometric
term must individually satisfy the balance equations in the interior of the
state space and further show that the geometric terms in an invariant measure
must have a pairwise-coupled structure. Finally, we show that at least one of
the coefficients in the linear combination must be negative.
