On the weak convergence of superpositions of point processes

Abstract

Let X be a a-compact, second countable space and Mr(X) the set of nonnegative Radon Borel measures on X, i.e., elements of ~r are defined on the Borel algebra ~ (X) and finite on compact sets. Denote by ~ ; the set of continuous functions X --~ R with compact support and endow Jr with the vague topology, generated by neighborhoods V(fi, ... ,f , , e, # )= {v ~ J/g(X); Iv f j -# f j l 0. Here and in the sequel #f=[. fd#. Harris [5, p. 111 ff.] has shown that probability measures on cylinder algebras in J/g(X) can be defined through projective systems satisfying obvious additivity and continuity conditions. In the present case his proof simplifies since compactness arguments may be used instead of completeness. Also the probability measures may be defined on the vague Borel algebra, M (~'). For any Borel probability measure on J//(X), define its characteristic functional [7] z(P) on ~K as the function