ARBITRARY FUNCTIONAL GLIVENKO-CANTELLI CLASSES AND APPLICATIONS TO DIFFERENT TYPES OF DEPENDENCE

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 ...

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...

Preservation 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...

On the geometry of Glivenko-Cantelli classes

Rademacher averages and phase transitions in Glivenko-Cantelli classes

We introduce a new parameter which may replace the fat-shattering dimension. Using this parameter we are able to provide improved complexity estimates for the agnostic learning problem with respect to any norm. Moreover, we show that if fat ( ) = ( ) then displays a clear phase transition which occurs at = 2. The phase transition appears in the sample complexity estimates, covering numbers esti...