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 P0L with respect to sets L of marks can all be used as starting point. When P is used, a modified, event-stationary version Q0L of P0L 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 P0L′, is taken as starting point, we can approximate another modified, event-stationary version of P0L by considering discrete-time Cesàro averages and a modified, time-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 P, P0L′, P0L′, and modified versions of them, and their connections with ergodicity conditions and long-run averages of Cesàro type.
