Gamma-type results and other related properties of Poisson processes

Families of Poisson processes defined on general state spaces and with the intensity measure scaled by a positive parameter are investigated. In particular, mean value relations with respect to the scale parameter are established and used to derive various Gamma-type results for certain geometric characteristics determined by finite subprocesses. In particular, the authors deduce Miles’ complementary theorem. Applications of the results within stochastic geometry and particularly for random tessellations are discussed.