Back to the Egg: Revising the definitions and basic properties of the Poisson and Poisson cluster point processes (a case study)

Motivated by our recent paper with A. Daletskii (“Poisson cluster measures: Quasi-invariance, integration by parts and equilibrium stochastic dynamics”, J. Func. Anal. 256 (2009), 432–478; doi: 10.1016/j.jfa.2008.10), and in particular by queries raised by the anonymous referee, we revise the definitions and basic properties of the Poisson and Poisson cluster point processes in a setting more general than usual. As a concrete result, we prove (novel) necessary and sufficient conditions for the Poisson cluster process to be proper (i.e., with a.s. no multiple or accumulation points).