  We consider a discrete Markov-additive process, that is a Markov chain on a
state space $\mathbb{Z}^d \times E$ with invariant jumps along the
$\mathbb{Z}^d$ component. In the case where the set $E$ is finite, we derive an
asymptotic equivalent of the Green function of the process, providing a new
proof of a result obtained by Dussaule in 2020. This result generalizes the
famous theorem of Ney and Spitzer of 1966, that deals with the sum of
independent and identically distributed random variables, to a spatially
non-homogeneous case.
  In this new proof, we generalize the arguments used in Woess's book Random
Walks on Infinite Graphs and Groups to prove Ney and Spitzer's theorem, that
consists in establishing an integral formula of the Green function from which
we get the asymptotic equivalent. To do so, we use techniques developed by
Babillot. In particular, we use dyadic splitting of integrals, a powerful
Fourier analysis tool that enables us to control the Fourier transform of a
function that has a singularity at the origin.
