Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry

In Annals of Probability, 2016

The Poisson point process is the first brick on which the stochastic geometry is built upon. Since we have a very well developed Malliavin calculus for this process, it is rather straightforward to apply the Stein-Malliavin-Dirichlet to this setting. This gives raise to this beautiful paper. We investigate several multi-points transformations of point processes which lead to a Poissonian limit. Once again, a cornerstone of the calculations is the Stein representation formula of the Rubsintein distance as an integral along the path of an Ornstein-Ulhenbeck type process. The magic here is that the whole machinery can also be used to prove convergence of some U-statistics without relying on some artificial hypothesis like having integer-valued marks in the limit.

L. Decreusefond
Professor of Probability