Glivenko-Cantelli for f-divergence

Abstract

We extend the celebrated Glivenko-Cantelli theorem, sometimes called the fundamental theorem of statistics, from its standard setting of total variation distance to all $f$-divergences. A key obstacle in this endeavor is to define $f$-divergence on a subcollection of a $σ$-algebra that forms a $π$-system but not a $σ$-subalgebra. This is a side contribution of our work. We will show that this notion of $f$-divergence on the $π$-system of rays preserves nearly all known properties of standard $f$-divergence, yields a novel integral representation of the Kolmogorov-Smirnov distance, and has a Glivenko-Cantelli theorem. We will also discuss the prospects of a Vapnik-Chervonenkis theory for $f$-divergence.