Uniform and universal Glivenko-Cantelli classes
A class \(\tilde F\) of measurable functions on a probability space is called a Glivenko-Cantelli class if the empirical measures \(P_ n\) converge to the true P uniformly over \(\tilde F\) almost surely. \(\tilde F\) is a universal Glivenko-Cantelli class if it is a Glivenko-Cantelli class for all laws P on a measurable space, and a uniform Glivenko- Cantelli class if the convergence is also uniform in P. The authors give general sufficient conditions for the Glivenko-Cantelli and universal Glivenko-Cantelli properties and examples to show that some stronger conditions are not necessary. The uniform Glivenko-Cantelli property is characterized, under measurability assumptions, by an entropy condition.
scientific article; zbMATH DE number 4211167
scientific article