## The Annals of Statistics

### Invariance Principle for Symmetric Statistics

#### Abstract

We derive invariance principles for processes associated with symmetric statistics of arbitrary order. Using a Poisson sample size, such processes can be viewed as functionals of a Poisson Point Process. Properly normalized, these functionals converge in distribution to functionals of a Gaussian random measure associated with the distribution of the observations. We thus obtain a natural description of the limiting process in terms of multiple Wiener integrals. The results are used to derive asymptotic expansions of processes arising from arbitrary square integrable $U$-statistics.

#### Article information

Source
Ann. Statist., Volume 12, Number 2 (1984), 483-496.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346501

Digital Object Identifier
doi:10.1214/aos/1176346501

Mathematical Reviews number (MathSciNet)
MR740907

Zentralblatt MATH identifier
0547.60039

JSTOR