Functional Convergence of Sequential U-processes with Size-Dependent Kernels

TL;DR

This paper establishes universal conditions under which sequences of size-dependent symmetric U-processes converge to a combination of time-changed Brownian motions, extending classical CLTs to infinite-dimensional and sample-size-dependent settings.

Contribution

It provides a set of analytic sufficient conditions for the weak convergence of size-dependent U-processes to Gaussian limits, generalizing previous CLTs and introducing new tools for analysis.

Findings

Derived universal convergence conditions for U-processes.

Extended CLTs to infinite-dimensional, size-dependent U-statistics.

Applied results to random geometric graphs and changepoint analysis.

Abstract

We consider sequences of $U$-processes based on symmetric kernels of a fixed order, that possibly depend on the sample size. Our main contribution is the derivation of a set of analytic sufficient conditions, under which the aforementioned $U$-processes weakly converge to a linear combination of time-changed independent Brownian motions. In view of the underlying symmetric structure, the involved time-changes and weights remarkably depend only on the order of the U-statistic, and have consequently a universal nature. Checking these sufficient conditions requires calculations that have roughly the same complexity as those involved in the computation of fourth moments and cumulants. As such, when applied to the degenerate case, our findings are infinite-dimensional extensions of the central limit theorems (CLTs) proved in de Jong (1990) and D\"obler and Peccati (2017). As important tools in our analysis, we exploit the multidimensional central limit theorems established in D\"obler and Peccati (2019) together with upper bounds on absolute moments of degenerate $U$-statistics by Ibragimov and Sharakhmetov (2002), and also prove some novel multiplication formulae for degenerate symmetric $U$-statistics -- allowing for different sample sizes -- that are of independent interest. We provide applications to random geometric graphs and to a class of $U$-statistics of order two, whose Gaussian fluctuations have been recently studied by Robins et al. (2016), in connection with quadratic estimators in non-parametric models. In particular, our application to random graphs yields a class of new functional central limit theorems for subgraph counting statistics, extending previous findings in the literature. Finally, some connections with invariance principles in changepoint analysis are established.