For $G=\mathbb{R}^d$ I know that a stationary point process $X$ either has 0 or infinitely many points, a.s. Daley and Vere-Jones refer to this as the 0-Infinity dichotomy. They hint that this fact is known in a more general setting. What is the most general setting for which the answer is known? Does it hold for all (locally compact second countable hausdorff topological) groups $G$? References appreciated.

Edit: I'm really interested in when the dichotomy DOES hold. If you have some example where it doesn't hold, please post as a comment unless you feel that your example really does answer my question.

Edit: Obviously $G$ must be non-compact for the dichotomy to hold, else a Poisson process will give a counterexample.