Ivanoff, B. Gail; Merzbach, Ely; Plante, Mathieu A compensator characterization of point processes on topological lattices. (English) Zbl 1127.60085 Electron. J. Probab. 12, 47-74 (2007). 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. Cited in 9 Documents MSC: 60K05 Renewal theory 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G48 Generalizations of martingales Keywords:multiparameter martingale; compensators; totally ordered point process; multiparameter renewal processes; Poisson process PDFBibTeX XMLCite \textit{B. G. Ivanoff} et al., Electron. J. Probab. 12, 47--74 (2007; Zbl 1127.60085) Full Text: DOI EuDML