Electronic Journal of Probability

Quantitative de Jong theorems in any dimension

Christian Döbler and Giovanni Peccati

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We develop a new quantitative approach to a multidimensional version of the well-known de Jong’s central limit theorem under optimal conditions, stating that a sequence of Hoeffding degenerate $U$-statistics whose fourth cumulants converge to zero satisfies a CLT, as soon as a Lindeberg-Feller type condition is verified. Our approach allows one to deduce explicit (and presumably optimal) Wasserstein bounds in the case of general $U$-statistics of arbitrary order $d\geq 1$. One of our main findings is that, for vectors of $U$-statistics satisfying de Jong’ s conditions and whose covariances admit a limit, componentwise convergence systematically implies joint convergence to Gaussian: this is the first instance in which such a phenomenon is described outside the frameworks of homogeneous chaoses and of diffusive Markov semigroups.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 2, 35 pp.

Received: 24 April 2016
Accepted: 15 December 2016
First available in Project Euclid: 5 January 2017

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Primary: 60F05: Central limit and other weak theorems 62E17: Approximations to distributions (nonasymptotic) 62E20: Asymptotic distribution theory

quantitative CLTs de Jong’s Theorem exchangeable pairs Hoeffding decomposition degenerate $U$-statistics multidimensional convergence Stein’s method

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Döbler, Christian; Peccati, Giovanni. Quantitative de Jong theorems in any dimension. Electron. J. Probab. 22 (2017), paper no. 2, 35 pp. doi:10.1214/16-EJP19. https://projecteuclid.org/euclid.ejp/1483585524

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