Ergodicity conditions and cesàro limit results for marked point processes

Abstract

In Palm theory it is very common to consider several distributions to describe the characteristics of the system. To study a stationary marked point process, the time-stationary distribution P and its event-stationary Palm distributions with respect to sets L of marks can all be used as starting point. When P is used, a modified, event-stationary version is defined as the limit of an obvious discrete-time Cesàro average. In a sense this modified Palm distribution is more natural than the ordinary one. When a Palm distribution is taken as starting point, we can approximate another modifiedevent-stationary version of by considering discrete-time Cesàro averages and a modifiedtime-stationary version QL of P by considering continuous-time Cesàro averages. These and other limit results are corollaries of uniform limit theorems for Cesàro averaged functionals. In essence, this paper presents a profound study of the relationship between and modified versions of them, and their connections with ergodicity conditions and long-run averages of Cesàro type.