Gert Nieuwenhuis

Copyright © 2001 Gert Nieuwenhuis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.