Improved criteria for distributional convergence of point processes
The author shows that for the weak convergence of point processes \(\xi_n\) on a locally compact second countable Hausdorff space to a simple point process \(\xi \), it is sufficient to assume that \(\Pr [\xi_n(U)=0]\to \Pr [\xi (u)=0]\) for all subsets \(U\) from a fixed `separating' class (separating open sets from included compact sets) and that \(\liminf_n\Pr [\xi_n(I)>1]\leq \Pr [\xi (I)>1]\) for all subsets \(I\) from a fixed (pre-)separating class of stochastic continuity sets for \(\xi\). The assumptions can be replaced by appropriate convergence conditions for the Laplace functionals. Similar results for the weak convergence of null-arrays of point processes to an infinite divisible point process are presented. These are improvements of results from an earlier book by the author [``Random measures'', 3rd, rev. and enl. ed. (1983; Zbl 0544.60053)], where an additional assumption assuring the tightness of the sequence of point process distributions was imposed. Similar results can be found in [\textit{D. J. Daley} and \textit{D. Vere-Jones}, ``An introduction to the theory of point processes'' (1988; Zbl 0657.60069), see Ch. 9, namely Proposition 9.1.IX].
scientific article; zbMATH DE number 1095920