Poisson Processes With Integrable Density

When the density of a Poisson process is integrable various expressions published in the literature are incorrect. This is especially the case of the probability distribution of the distance between an origin and the following points of the process. The first purpose of this paper is to explain why the integrability of the density changes the situation. The second is to discuss various consequences of this fact on the probability distribution of random variables extracted from the process. Computer experiments are presented and are in excellent agreement with theoretical results. Some extensions of the same problem concerning renewal processes are discussed