Quantitative de Jong theorems in any dimension

Abstract

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.