Functionals of Poisson Processes and Applications

In this chapter, we provide the foundations of a versatile and powerful method for treating Poisson functionals in general state spaces that leads to representations of variances and bounds on the Wasserstein distance from a Gaussian distribution. This general theory, which has essentially evolved in the past 20 years, allows us to improve earlier results and opens the way to new applications. The first section focuses on Poisson functionals with a finite chaos expansion. Earlier results are recovered. Section 2 extends the framework and treats general Poisson functionals. In Section 3, we turn to bounds for the Wasserstein distance of general Poisson functionals and a standard normal distribution. An application of the previously developed techniques is presented in the final section, where we investigate the analogue of intersection processes of Poisson hyperplane processes in hyperbolic space. Unexpected phenomena and new asymptotic scenarios arise.