Invariance Principle for Symmetric Statistics

The Annals of Statistics · 1984 · Volume 12 · Issue 2

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.

Keywords

$U$-statistics, 60F17, 60G99, 60K99, 62E20, 62G05, Hermite polynomials, invariance principle, multiple Wiener integral, symmetric statistics

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