The multivariate functional de Jong CLT

TL;DR

This paper establishes a multivariate functional central limit theorem for sequences of Hoeffding-degenerate U-statistics, demonstrating weak convergence of empirical processes under specific cumulant and Lindeberg conditions, and extends the universality of Wiener chaos to the functional setting.

Contribution

It introduces a multivariate functional CLT for U-statistics and extends the universality of Wiener chaos to the functional level.

Findings

Weak convergence of empirical processes of U-statistics under cumulant conditions

Extension of Wiener chaos universality to functional processes

Conditions involving fourth cumulants and Lindeberg-type criteria

Abstract

We prove a multivariate functional version of de Jong's CLT (1990) yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding empirical processes on $[0,1]$ weakly converge in the Skorohod space as soon as their fourth cumulants in $t=1$ vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified. As an application, we lift to the functional level the `universality of Wiener chaos' phenomenon first observed in Nourdin, Peccati and Reinert (2010).