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A compensator characterization of point processes on topological lattices. (English) Zbl 1127.60085

Summary: We resolve the longstanding question of how to define the compensator of a point process on a general partially ordered set in such a way that the compensator exists, is unique, and characterizes the law of the process. We define a family of one-parameter compensators and prove that this family is unique in some sense and characterizes the finite dimensional distributions of a totally ordered point process. This result can then be applied to a general point process since we prove that such a process can be embedded into a totally ordered point process on a larger space. We present some examples, including the partial sum multiparameter process, single line point processes, multiparameter renewal processes, and obtain a new characterization of the two-parameter Poisson process.

MSC:

60K05 Renewal theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G48 Generalizations of martingales
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