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 P. 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 P and its DPD. Under P, 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.
