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

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.

MSC:

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