Glivenko-Cantelli for $f$-divergence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917966391541760 |
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| author | Wang, Haoming Lim, Lek-Heng |
| author_facet | Wang, Haoming Lim, Lek-Heng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_17355 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Glivenko-Cantelli for $f$-divergence Wang, Haoming Lim, Lek-Heng Statistics Theory Machine Learning 60B10, 60F15, 60F25 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. |
| title | Glivenko-Cantelli for $f$-divergence |
| topic | Statistics Theory Machine Learning 60B10, 60F15, 60F25 |
| url | https://arxiv.org/abs/2503.17355 |