Symmetric Statistics, Poisson Point Processes, and Multiple Wiener Integrals

Abstract

The asymptotic behaviour of symmetric statistics of arbitrary order is studied. As an application we describe all limit distributions of square integrable $U$-statistics. We use as a tool a randomization of the sample size. A sample of Poisson size $N_\lambda$ with $EN_\lambda = \lambda$ can be interpreted as a Poisson point process with intensity $\lambda$, and randomized symmetric statistics are its functionals. As $\lambda \rightarrow \infty$, the probability distribution of these functionals tend to the distribution of multiple Wiener integrals. This can be considered as a stronger form of the following well-known fact: properly normalized, a Poisson point process with intensity $\lambda$ approaches a Gaussian random measure, as $\lambda \rightarrow \infty$.