  In this paper, we prove that ergodic point processes with moments of all
orders, driven by particular infinite measure preserving transformations, have
to be a superposition of shifted Poisson processes. This rigidity result has a
lot of implications in terms of joining and disjointness for the corresponding
Poisson suspension. In particular, we prove that its ergodic self-joinings are
Poisson joinings, which provides an analog, in the Poissonian context, of the
GAG property for Gaussian dynamical systems.
