February 2022 Functional convergence of sequential U-processes with size-dependent kernels
Christian Döbler, Mikołaj J. Kasprzak, Giovanni Peccati
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Ann. Appl. Probab. 32(1): 551-601 (February 2022). DOI: 10.1214/21-AAP1688

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 (J. Multivariate Anal. 34 (1990) 275–289) and Döbler and Peccati (Electron. J. Probab. 22 (2017) Paper No. 2). As important tools in our analysis, we exploit the multidimensional central limit theorems established in Döbler and Peccati (Electron. J. Probab. 24 (2019) Paper No. 5) together with upper bounds on absolute moments of degenerate U-statistics by Ibragimov and Sharakhmetov (Studia Sci. Math. Hungar. 39 (2002) 251–275), 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. (Stochastic Process. Appl. 126 (2016) 3733–3759), in connection with quadratic estimators in nonparametric 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.

Funding Statement

The research developed in this paper is supported by the FNR Grant FoRGES (R-AGR-3376-10) at Luxembourg University.

Acknowledgements

We thank two anonymous referees for several constructive remarks. We are grateful to Yannick Baraud, Andrew D. Barbour, Omar El-Dakkak and Ivan Nourdin for useful discussions.

Citation

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Christian Döbler. Mikołaj J. Kasprzak. Giovanni Peccati. "Functional convergence of sequential U-processes with size-dependent kernels." Ann. Appl. Probab. 32 (1) 551 - 601, February 2022. https://doi.org/10.1214/21-AAP1688

Information

Received: 1 December 2019; Revised: 1 October 2020; Published: February 2022
First available in Project Euclid: 27 February 2022

MathSciNet: MR4386536
Digital Object Identifier: 10.1214/21-AAP1688

Subjects:
Primary: 60F17
Secondary: 60D05 , 62G20

Keywords: Asymptotic properties of estimators , contractions , Functional limit theorems , Hoeffding decompositions , product formulae , Random geometric graphs , Stein’s method , U-statistics

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.32 • No. 1 • February 2022
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