A compensator characterization of point processes on toplogical lattices

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