Laws of Large Numbers for Information Resolution
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915286016327680 |
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| author | Raban, Daniel |
| author_facet | Raban, Daniel |
| contents | Laws of large numbers establish asymptotic guarantees for recovering features of a probability distribution using independent samples. We introduce a framework for proving analogous results for recovery of the $σ$-field of a probability space, interpreted as information resolution--the granularity of measurable events given by comparison to our samples. Our main results show that, under iid sampling, the Borel $σ$-field in $\mathbb R^d$ and in more general metric spaces can be recovered in the strongest possible mode of convergence. We also derive finite-sample $L^1$ bounds for uniform convergence of $σ$-fields on $[0,1]^d$.
We illustrate the use of our framework with two applications: constructing randomized solutions to the Skorokhod embedding problem, and analyzing the loss of variants of random forests for regression. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07958 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Laws of Large Numbers for Information Resolution Raban, Daniel Probability Statistics Theory 60F15, 60A10 (Primary) 60G40 62G08 (Secondary) Laws of large numbers establish asymptotic guarantees for recovering features of a probability distribution using independent samples. We introduce a framework for proving analogous results for recovery of the $σ$-field of a probability space, interpreted as information resolution--the granularity of measurable events given by comparison to our samples. Our main results show that, under iid sampling, the Borel $σ$-field in $\mathbb R^d$ and in more general metric spaces can be recovered in the strongest possible mode of convergence. We also derive finite-sample $L^1$ bounds for uniform convergence of $σ$-fields on $[0,1]^d$. We illustrate the use of our framework with two applications: constructing randomized solutions to the Skorokhod embedding problem, and analyzing the loss of variants of random forests for regression. |
| title | Laws of Large Numbers for Information Resolution |
| topic | Probability Statistics Theory 60F15, 60A10 (Primary) 60G40 62G08 (Secondary) |
| url | https://arxiv.org/abs/2505.07958 |