  We consider one-dependent random walks on $\mathbb{Z}^d$ in random
hypergeometric environment for $d\ge 3$. These are memory-one walks in a large
class of environments parameterized by positive weights on directed edges and
on pairs of directed edges which includes the class of Dirichlet environments
as a special case. We show that the walk is a.s. transient for any choice of
the parameters, and moreover that the return time has some finite positive
moment. We then give a characterization for the existence of an invariant
measure for the process from the point of view of the walker which is
absolutely continuous with respect to the initial distribution on the
environment in terms of a function $\kappa$ of the initial weights. These
results generalize [Sab11] and [Sab13] on random walks in Dirichlet
environment. It turns out that $\kappa$ coincides with the corresponding
parameter in the Dirichlet case, and so in particular the existence of such
invariant measures is independent of the weights on pairs of directed edges,
and determined solely by the weights on directed edges.
