Convergence in distribution of point processes on Polish spaces to a simple limit

Let ξ,ξ1,ξ2,… be a sequence of point processes on a complete and separable metric space (S,d) with ξ simple. We assume that P{ξnB=0}→P{ξB=0} and lim supn→∞P{ξnB>1}≤P{ξB>1} for all B in some suitable class B, and show that this assumption determines if the sequence {ξn} converges in distribution to ξ. This is an extension to general Polish spaces of the weak convergence theory for point processes on locally compact Polish spaces found in Kallenberg (1996).