# Glivenko-Cantelli for f-divergence

arXiv: https://arxiv.org/abs/2503.17355 · HTML: https://mlsanity.com/paper/2503.17355/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 53 references that are in the map.
- First posted on arXiv 21 Mar 2025.

## Builds on (most cited first, 1 of 1)

- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)

## Abstract (arXiv)

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.