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

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 Döbler and Peccati (2017). As important tools in our analysis, we exploit the multidimensional central limit theorems established in Döbler 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.

Asymptotic error distribution for the Riemann approximation of integrals driven by fractional Brownian motion

“…Let (X n ) ⊂ C([0, T ]) be a sequence of stochastic processes that can be decomposed as X n t = A n t + C n t , with A n , C n ∈ D([0, T ]) and, for some α, β, K > 0, (20) E Proof. In [5], it is proved that the sequence (A n ) is tight in D([0, T ]). Moreover, the sequence (C n ) is also tight in D([0, T ]), so is (X n ) as a sum of tight sequences.…”

Stein's method of exchangeable pairs in multivariate functional approximations

“…Moreover, since one main class of applications in the present paper involves weighted U -processes, it is worthwhile to compare our results and their applicability to those of [DKP19]. Firstly, as mentioned above, the paper [DKP19] focuses on Gaussian limit theorems for symmetric U -processes, which constitute a narrower class than the weighted U -processes considered in the present work.…”

“…In the context of these three groups of references and the present paper, we also mention the recent paper [DKP19] which, although not relying on functional approximation by Stein's method, provides functional limit theorems for the class of (degenerate and non-degenerate) symmetric U -processes with a kernel that may depend on the sample size n. Since it implicitly relies on a multivariate Gaussian limit theorem derived by Stein's method from [DP19], it is also naturally related to Stein's method.…”

“…Firstly, as mentioned above, the paper [DKP19] focuses on Gaussian limit theorems for symmetric U -processes, which constitute a narrower class than the weighted U -processes considered in the present work. Moreover, thanks to the finite-dimensional convergence results from [DP19], the conditions for convergence from [DKP19] are phrased in term of L 2 -norms of contraction kernels and, as such, can be considered as fourth moment conditions. Contrarily, as can be seen from the bounds and proofs of Section 5, the bounds and conditions in the present paper involve third moment quantities.…”

“…As a concrete example, in Subsection 5.6, we provide a bound for a Gaussian approximation of a process that is defined as a vector of success runs of different lengths. For functional limit theorems involving the class of symmetric, degenerate U -processes, we refer the reader to the recent paper [DKP19]. Moreover, we remark that, even in the univariate case of weighted U -statistics, the literature about limit theory for these random quantities is quite restricted.…”