Weak convergence of set-indexed point processes and Poisson processes. (English) Zbl 0923.60003

Ivanoff, B. Gail; Lin, Yan-Xia; Merzbach, Ely

Theory Probab. Math. Stat. 55, 79-91 (1997) and Teor. Jmovirn. Mat. Stat. 55, 77-89 (1996).

Let \({\mathcal A}\) be a collection of closed subsets of a compact topological space \(T\) and \(D({\mathcal A})\) be a space of set-indexed functions which are outer continuous with inner limits. The separable metric \(D_H\) on \(D({\mathcal A})\) is defined by the convergence of graphs with respect to Hausdorff distance. If \(T=[0,1]\) and \({\mathcal A}=\{[0,x], 0\leq x\leq 1\}\), then \(D({\mathcal A})\) is homeomorphic with \(D[0,1]\) endowed with Skorokhod \(J_2\) topology. The concepts of set-indexed martingale, submartingale and compensator are introduced. A criterion for weak convergence for a sequence of set-indexed simple point processes is obtained. As a consequence it is proved that weak convergence of a point process to a Poisson process is implied by the uniform pointwise convergence of the respective compensators to a deterministic diffuse measure.

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60B10	Convergence of probability measures
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60G48	Generalizations of martingales

Keywords:

set-indexed function; Hausdorff distance; weak convergence; point measure; point process; Poisson process; compensator