FUNCTIONAL CONVERGENCE OF SEQUENTIAL U-PROCESSES WITH SIZE-DEPENDENT KERNELS

• 成果类型: Article
• 署名作者: Doebler, Christian; Kasprzak, Mikolaj J.; Peccati, Giovanni
• 署名单位: Heinrich Heine University Dusseldorf; University of Luxembourg
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1688
• 发表日期: 2022
• 页码: 551-601
• 关键词: limit-theorems; integral functionals; Asymptotic Normality; statistics; approximations; MULTIVARIATE; density

摘要

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 Dobler 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 Dobler 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.