منابع مشابه
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 ...
متن کاملPreservation 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...
متن کاملEmpirical 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 ...
متن کاملPreservation Theorems for Glivenko-cantelli and Uniform Glivenko-cantelli Classes Aad Van Der Vaart and Jon
We show that the P Glivenko property of classes of functions F1; : : : ;Fk is preserved by a continuous function ' from R k to R in the sense that the new class of functions x! '(f1(x); : : : ; fk(x)); fi 2 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 pro...
متن کاملThe Glivenko Cantelli Theorem and its Generalizations
In this note we will study upper bounds of random variables of the type sup A∈A |ν n (A) − ν(A)| , where A is a class of sets that needs to fulll certain assumptions. These bounds are important tools in the analysis of learning processes and probabilistic theories of pattern recognition. The presentation given here is based on [DGL96].
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ژورنال
عنوان ژورنال: The Annals of Probability
سال: 1987
ISSN: 0091-1798
DOI: 10.1214/aop/1176992069