Uniformity in the Underlying Distribution

Abstract

The previous chapters present empirical laws of large numbers and central limit theorems for observations from a fixed underlying distribution P. Many of the sufficient conditions given there are actually satisfied by very large classes of underlying measures; typically, the only limitation is finiteness of some appropriate moment of the envelope function. For instance, classes satisfying the uniform entropy condition are, up to measurability, Glivenko-Cantelli or Donsker for all P with P*F < ∞ or P*F ² < ∞, respectively. In particular, many bounded classes of functions are universally Donsker: Donsker for every probability measure on the sample space.