Stein, Rubinstein, Malliavin, Poisson
One of many advantages of the Stein’s method is its versatility. It can be applied to prove convergence to Gaussian random variables or processes as well as to Poisson random variables or point processes. In this new paper, we investigate this last aspect showing that some multi-points transformation of some point processes 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.