Quantitative de Jong theorems in any dimension

TL;DR

This paper introduces a new quantitative method for multidimensional de Jong's CLT, providing explicit Berry-Esseen bounds and establishing joint Gaussian convergence from componentwise convergence for vectors of U-statistics.

Contribution

It develops a novel approach to multidimensional de Jong's CLT, deriving explicit bounds and demonstrating joint convergence under broad conditions, extending beyond previous frameworks.

Findings

Explicit Berry-Esseen bounds for general U-statistics.

Componentwise convergence implies joint Gaussian convergence.

First demonstration of this phenomenon outside homogeneous chaoses and Markov semigroups.

Abstract

We develop a new quantitative approach to a multidimensional version of the well-known {\it 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) Berry-Esseen bounds in the case of general $U$-statistics of arbitrary order $d\geq1$. 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.