The multivariate functional de Jong CLT

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).

24 pages, submitted version, to appear in Probability Theory and Related Fields (2022), published version available at https://link.springer.com/article/10.1007/s00440-022-01114-3

References in corpus (5)

• Proofs of the martingale FCLT
• Fourth moment theorems on the Poisson space in any dimension
• Stein's method for multivariate Brownian approximations of sums under dependence
• Functional approximations with Stein's method of exchangeable pairs
• Stein's method of exchangeable pairs in multivariate functional approximations