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 $\lambda \rightarrow \infty$.
Publié le : 1983-09-14
Classification:
Symmetric statistic,
Multiple Wiener Integral,
Poisson Point Process,
$U$-statistic,
60F05,
62E10,
62G05,
60G15,
60G55
@article{1176346241,
author = {Dynkin, E. B. and Mandelbaum, A.},
title = {Symmetric Statistics, Poisson Point Processes, and Multiple Wiener Integrals},
journal = {Ann. Statist.},
volume = {11},
number = {1},
year = {1983},
pages = { 739-745},
language = {en},
url = {http://dml.mathdoc.fr/item/1176346241}
}
Dynkin, E. B.; Mandelbaum, A. Symmetric Statistics, Poisson Point Processes, and Multiple Wiener Integrals. Ann. Statist., Tome 11 (1983) no. 1, pp. 739-745. http://gdmltest.u-ga.fr/item/1176346241/