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

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.

Stein's method for Brownian approximations