ABSTRACT
A point process is a strictly increasing sequence of real numbers, which does not have a finite limit point. Although the majority of applications of point processes refer to sequences of time points, there are other interpretations as well. The most important recurrent point processes are homogeneous Poisson processess and renewal processes. According to the underlying point process, there are e.g. compound Poisson processes and compound renewal processes. In the theory of stochastic processes, and maybe even more in its applications, the homogeneous Poisson process is just as popular as the exponential distribution in probability theory. The relationship between homogeneous Poisson processes and the uniform distribution proved in this theorem motivates the common phrase that a homogeneous Poisson process is a purely random process. Random point processes are key tools for quantifying the financial risk in virtually all branches of industry. This section uses the terminology for analyzing the financial risk in the insurance industry.
