## The Annals of Statistics

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

### Symmetric Statistics, Poisson Point Processes, and Multiple Wiener Integrals

E. B. Dynkin and A. Mandelbaum

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 62E10: Characterization and structure theory 62G05: Estimation 60G15: Gaussian processes 60G55: Point processes

**Keywords**

Symmetric statistic Multiple Wiener Integral Poisson Point Process $U$-statistic

#### Citation

Dynkin, E. B.; Mandelbaum, A. Symmetric Statistics, Poisson Point Processes, and Multiple Wiener Integrals. Ann. Statist. 11 (1983), no. 3, 739--745. doi:10.1214/aos/1176346241. https://projecteuclid.org/euclid.aos/1176346241