Asymptotic mean stationarity and absolute continuity of point process distributions

Abstract

This paper relates – for point processes $\Phi$ on $\mathbb{R}$ – two types of asymptotic mean stationarity (AMS) properties and several absolute continuity results for the common probability measures emerging from point process theory. It is proven that $\Phi$ is AMS under the time-shifts if and only if it is AMS under the event-shifts. The consequences for the accompanying two types of ergodic theorem are considered. Furthermore, the AMS properties are equivalent or closely related to several absolute continuity results. Thus, the class of AMS point processes is characterized in several ways. Many results from stationary point process theory are generalized for AMS point processes. To obtain these results, we first use Campbell’s equation to rewrite the well-known Palm relationship for general nonstationary point processes into expressions which resemble results from stationary point process theory.