Open Access
2017 Quantitative de Jong theorems in any dimension
Christian Döbler, Giovanni Peccati
Electron. J. Probab. 22: 1-35 (2017). DOI: 10.1214/16-EJP19


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.


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Christian Döbler. Giovanni Peccati. "Quantitative de Jong theorems in any dimension." Electron. J. Probab. 22 1 - 35, 2017.


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

zbMATH: 1357.60023
MathSciNet: MR3613695
Digital Object Identifier: 10.1214/16-EJP19

Primary: 60F05 , 62E17 , 62E20

Keywords: de Jong’s Theorem , degenerate $U$-statistics , Exchangeable pairs , Hoeffding decomposition , multidimensional convergence , quantitative CLTs , Stein’s method

Vol.22 • 2017
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