Two-step procedures in Palm theory
Random time changes (RTCs) are right-continuous and non-decreasing random functions passing the zero-level at 0. The behavior of such systems can be studied from a randomly chosen time-point and from a randomly chosen level. From the first point of view, the probability characteristics are described by the time-stationary distribution . From the second point of view, the detailed Palm distribution (DPD) is the ruling probability mechanism. The main topic of the present paper is a relationship between and its DPD. Under , the origin falls in a continuous part of the graph. Under the DPD, there are two typical situations: the origin lies in a jump-part of the extended graph or it lies in a continuous part. These observations lead to two conditional DPDs. We derive two-step procedures, which bridge the gaps between the several distributions. One step concerns the application of a shift, the second is just a change of measure arranged by a weight-function. The procedures are used to derive local characterization results for the distributions of Palm type. We also consider simulation applications. For instance, a procedure is mentioned to generate a simulation of the RTC as seen from a randomly chosen level in a jump-part when starting with simulations from a randomly chosen time-point. The point process with batch-arrivals is often used as an application.