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September, 1983 Symmetric Statistics, Poisson Point Processes, and Multiple Wiener Integrals
E. B. Dynkin, A. Mandelbaum
Ann. Statist. 11(3): 739-745 (September, 1983). DOI: 10.1214/aos/1176346241


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 $\lambda \rightarrow \infty$.


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E. B. Dynkin. A. Mandelbaum. "Symmetric Statistics, Poisson Point Processes, and Multiple Wiener Integrals." Ann. Statist. 11 (3) 739 - 745, September, 1983.


Published: September, 1983
First available in Project Euclid: 12 April 2007

zbMATH: 0518.60050
MathSciNet: MR707925
Digital Object Identifier: 10.1214/aos/1176346241

Primary: 60F05
Secondary: 60G15, 60G55, 62E10, 62G05

Rights: Copyright © 1983 Institute of Mathematical Statistics


Vol.11 • No. 3 • September, 1983
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