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

Abstract

Let ξ , ξ 1 , ξ 2 , … be a sequence of point processes on a complete and separable metric space ( S , d ) with ξ simple. We assume that P { ξ n B = 0 } → P { ξ B = 0 } and lim sup n → ∞ P { ξ n B > 1 } ≤ P { ξ B > 1 } for all B in some suitable class B , and show that this assumption determines if the sequence { ξ n } 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).