Central limit theorem for multiple integrals with respect to the empirical process

Joint work with Eustasio del Barrio. Article publicated in Statistics and Probability Letters, vol. 79(2), p. 188-195, January 2009.
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In that article, we give some results of weak convergence of multiple integrals with respect to the empirical process. We consider objects of type
$$J_{n,m}(h)=\int’h(x_1, \dots, x_m)d\mathbb{G}_n(x_1)\dots \mathbb{G}_n(x_m),$$
where h is a symmetric real-valued square integrable function of m variables. $$X_1, \dots, X_n$$ is a P-distributed i.i.d. sample, and $$\mathbb{P}_n=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$$ and $$\mathbb{G}_n=\sqrt{n}(\mathbb{P}_n-P)$$ are respectively the associated empirical measure and the empirical process. $$\int’$$ is the integral where the integration on the diagonal has been omitted. We include the case of non degenerate kernels with respect to the underlying distribution. Our results are related to earlier results on U-statistics. We introduce a stochastic integral with respect to the Brownian bridge which allows us to express the limit in a unified way in the degenerate and non degenerate cases. Using the multiple integral with respect to the empirical process has an advantage with respect to using U-statistics: the Central Limit Theorem we obtain is simpler. It does not involve the degeneracy of the kernel and the limit is expressed in a precise way.