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

Let xi, xi(1), xi(2), ... be a sequence of point processes on a complete and separable metric space (S, d) with xi simple. We assume that P{xi(n)B = 0} -> P{xi B = 0} and lim sup(n ->infinity) P{xi(n)B > 1} <= P{xi B > 1} for all B in some suitable class 2, and show that this assumption determines if the sequence {xi(n)} converges in distribution to xi. 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). (C) 2011 Elsevier B.V. All rights reserved.