## The Annals of Statistics

### 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 $\lambda \rightarrow \infty$.

#### Article information

Source
Ann. Statist., Volume 11, Number 3 (1983), 739-745.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346241

Digital Object Identifier
doi:10.1214/aos/1176346241

Mathematical Reviews number (MathSciNet)
MR707925

Zentralblatt MATH identifier
0518.60050

JSTOR