Weak convergence and Glivenko-Cantelli results for empirical processes of u-statistic structure

Empirical Processes: Glivenko–Cantelli Theorems

The reader is referred to Chapter 1.6 of Wellner’s Torgnon notes, Chapter ??? of VDVW and Chapter 8.3 of Kosorok. First, a theorem using bracketing entropy. Let (F , ‖ ‖) be a subset of a normed space of real functions f : X → R. Given real functions l and u on X (but not necessarily in F), the bracket [l, u] is defined as the set of all functions f ∈ F satisfying l ≤ f ≤ u. The functions l, u ...

Moment Inequalities for Supremum of Empirical Processes of‎ ‎U-Statistic Structure and Application to Density Estimation

We derive moment inequalities for the supremum of empirical processes of U-Statistic structure and give application to kernel type density  estimation ‎and estimation of the distribution function for functions of observations.  

Preservation Theorems for Glivenko-Cantelli and Uniform Glivenko-Cantelli Classes

We show that the P−Glivenko property of classes of functions F1, . . . ,Fk is preserved by a continuous function φ from R to R in the sense that the new class of functions x → φ(f1(x), . . . , fk(x)), fi ∈ Fi, i = 1, . . . , k is again a Glivenko-Cantelli class of functions if it has an integrable envelope. We also prove an analogous result for preservation of the uniform Glivenko-Cantelli prop...

Donsker and Glivenko-Cantelli theorems for a class of processes generalizing the empirical process

Uniform Glivenko–Cantelli Classes

A class of sets, or functions, is said to be P–Glivenko–Cantelli if the empirical measure Pn converges in some sense to the true measure, P , as n → ∞, uniformly over the class of sets or functions. Thus, the notions of Glivenko–Cantelli, and likewise uniform Glivenko–Cantelli are for the most part qualitative assessments of how “well–behaved” a collection of sets or functions is, in the sense ...