Glivenko-Cantelli for $f$-divergence

• URL: http://arxiv.org/abs/2503.17355v2
• Date: Mon, 24 Mar 2025 13:10:28 GMT
• Title: Glivenko-Cantelli for $f$-divergence
• Abstract summary: We extend the celebrated Glivenko-Cantelli theorem, sometimes called the fundamental theorem of statistics.<n>A key obstacle is to define $f$-divergence on a subcollection of a $sigma$-algebra that forms a $pi$-system but not a $sigma$-subalgebra.<n>We will show that this notion of $f$-divergence on the $pi$-system of rays preserves nearly all known properties of standard $f$-divergence.
• Score: 5.478764356647437
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\sigma$-algebra that forms a $\pi$-system but not a $\sigma$-subalgebra. This is a side contribution of our work. We will show that this notion of $f$-divergence on the $\pi$-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.

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