The Annals of Statistics

Invariance Principle for Symmetric Statistics

Avi Mandelbaum and Murad S. Taqqu

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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

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

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 62E20: Asymptotic distribution theory 62G05: Estimation 60G99: None of the above, but in this section 60K99: None of the above, but in this section

Symmetric statistics invariance principle multiple Wiener integral $U$-statistics Hermite polynomials


Mandelbaum, Avi; Taqqu, Murad S. Invariance Principle for Symmetric Statistics. Ann. Statist. 12 (1984), no. 2, 483--496. doi:10.1214/aos/1176346501.

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