2016/03/02 by Döbler, Christian, Peccati, Giovanni · 1 citation
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1603.00804
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≥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.