On Stationarity and Superposition of Point Processes

Abstract

This paper applies ideas from random set theory to simple point processes. We show stationarity of the hitting distributions suffices for the strict stationarity of a simple point process, but that in general all forms of stationarity differ. We compare and contrast the superposition operations of summation for random measures and union for random sets, specialized to point processes. Finally we consider completely random sets and their factors.