Limit theorems for a class of processes generalizing the U-empirical process

Abstract

In this paper, we develop theory and tools for studying U-processes, a natural higher-order generalization of the empirical processes. We introduce a class of random discrete U-measures that generalize the empirical U-measure. We establish a Glivenko-Cantelli and a Donsker theorem under conditions on entropy numbers prevalent in the theory of empirical processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The uniform limit theorems discussed in this paper are key tools for many further developments involving empirical process techniques. Our results are applied to prove the asymptotic normality of Liu's simplicial median. We conclude this paper by extending Anscombe's central limit theorem to encompass randomly stopped U-processes, building upon its application to randomly stopped sums of independent random variables.

Keywords:

• Empirical processes
• random measures
• U-processes

2000 Mathematics Subject Classifications:

• 60F17
• 62E17

Acknowledgments

The authors express their gratitude to the Editor-in-Chief Prof. Saul Jacka, an Associate Editor, and three referees for their invaluable feedback and thorough review. Their constructive comments and meticulous reading significantly enhanced the original work, leading to a more refined and focused presentation. The first author is extremely honored to dedicate this work to his mother, who is an exemplary role model for him.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding

This research received no external funding.

Data availability statement

Not applicable.

CRediT author statement

Inass Soukarieh: Methodology, Investigation, Writing – Original Draft, Writing – Review & Editing.

Salim Bouzebda: Conceptualization, Methodology, Investigation, Writing – Original Draft, Writing – Review & Editing, Supervision.

Both authors contributed equally to this work.

Notes

1 ⊕ is the orthogonal sum in $L_2({\mathcal{X}}^{\mathrm{\infty }},{\mathcal{A}}^{\mathrm{\infty }},{\mathbb{P}}^{\mathrm{\infty }})$.

2 In other words, $({\xi }_1,\dots ,{\xi }_n){=}_d({\xi }_{\pi \left(1\right)},\dots ,{\xi }_{\pi \left(n\right)})$ for any permutation π of $\{1,\dots ,n\}.$

3 The bracketing number, denoted by ${\mathcal{N}}_{\left[\right]}(u,\mathcal{F},d)$, where d is any pseudonorm on $\mathcal{F}$, describes then the geometric complexity of such a function class $\mathcal{F}$. It is defined as $\begin{array}{rl}{\mathcal{N}}_{\left[\right]}(u,\mathcal{F},d)& =min\{m:\mathrm{\exists }{f}_1,\dots ,f_m\mathrm{and}\mathrm{\exists }{b}_1,\dots ,b_m\mathrm{with}\underset{1⩽i⩽m}{max}d\left(b_i\right)⩽u\\ & \text{such that}\mathrm{\forall }f\in \mathcal{F},\mathrm{\exists i}\in \{1,\dots ,m\}\text{with}|f-f_i|⩽b_i\}.\end{array}$

4 The interest of Poissonization relies on the nice properties of Poisson processes, namely the independence of their increments and the behaviour of their moments. These properties considerably simplify calculations, to be more precise, if η is a Poisson random variable independent of the i.i.d. sequence $X_i,i\in \mathbb{N}$, $X_0=0$, and if $A_k,k\in \mathbb{N}$, are disjoint measurable sets, then the processes $\sum _{i=0}^{\eta }\mathbb{1}(X_i\in A_k){\delta }_{X_i},k=1,2,\dots$, are independent.

5 Let $\mathcal{P}\left(M\right)$ denote the collection of all probability measures on the measurable space $(M,\mathcal{B}(M\left)\right)$. The Lévy–Prokhorov metric $d_{\mathrm{LP}}:\mathcal{P}(M{)}^2\to [0,+\mathrm{\infty })$ is defined by setting the distance between two probability measures $d_{\mathrm{LP}}(\mu ,\nu ):=inf\{ϵ>0|\mu (A)\le \nu (A^{ϵ})+ϵ\mathrm{and}\nu (A)\le \mu (A^{ϵ})+ϵ\mathrm{for}\mathrm{all}A\in \mathcal{B}(M\left)\right\}.$

6 ⊕ is the orthogonal sum in $L_2(E^{\mathrm{\infty }},{\mathcal{E}}^{\mathrm{\infty }},P^{\mathrm{\infty }})$

7 The detailed theory of Cesàro convergence is discussed in Hardy's classic textbook [71]