Empirical Measures and Strong Laws of Large Numbers in Categorical Probability

[2503.21576] Empirical Measures and Strong Laws of Large Numbers in Categorical Probability Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2503.21576 (math) [Submitted on 27 Mar 2025 ( v1 ), last revised 4 May 2026 (this version, v4)] Title: Empirical Measures and Strong Laws of Large Numbers in Categorical Probability Authors: Tobias Fritz , Tomáš Gonda , Antonio Lorenzin , Paolo Perrone , Areeb Shah Mohammed View a PDF of the paper titled Empirical Measures and Strong Laws of Large Numbers in Categorical Probability, by Tobias Fritz and 4 other authors View PDF HTML (experimental) Abstract: The Glivenko--Cantelli theorem is a uniform version of the strong law of large numbers. It states that for every IID sequence of random variables, the empirical measure converges to the underlying distribution (in the sense of uniform convergence of the CDF). In this work, we provide tools to study such limits of empirical measures in categorical probability. We propose two axioms, namely permutation invariance and empirical adequacy, that a morphism of type $X^{\mathbb{N}} \to X$ should satisfy to be interpretable as taking an infinite sequence as input and producing a sample from its empirical measure as output. Since not all sequences have a well-defined empirical measure, such \emph{empirical sampling morphisms} live in quasi-Markov categories, which, unlike Markov categories, allow for partial morphisms. Given an empirical sampling morphism and a few other properties, we prove representability as well as abstract versions of the de Finetti theorem, the Glivenko--Cantelli theorem and the strong law of large numbers. We provide several concrete constructions of empirical sampling morphisms as partially defined Markov kernels on standard Borel spaces. Instantiating our abstract results then recovers the standard Glivenko--Cantelli theorem and the strong law of large numbers for random variables with finite first moment. Our work thus provides a joint proof of these two theorems in conjunction with the de Finetti theorem from first principles. Subjects: Probability (math.PR) ; Logic in Computer Science (cs.LO); Category Theory (math.CT); Statistics Theory (math.ST) Cite as: arXiv:2503.21576 [math.PR] (or arXiv:2503.21576v4 [math.PR] for this version) https://doi.org/10.48550/arXiv.2503.21576 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Logical Methods in Computer Science, Volume 22, Issue 2 (May 5, 2026) lmcs:15844 Related DOI : https://doi.org/10.46298/lmcs-22%282%3A13%292026 Focus to learn more DOI(s) linking to related resources Submission history From: Tobias Fritz [ view email ] [via LMCS proxy] [v1] Thu, 27 Mar 2025 14:56:26 UTC (80 KB) [v2] Fri, 20 Feb 2026 20:15:08 UTC (125 KB) [v3] Sat, 11 Apr 2026 06:17:35 UTC (93 KB) [v4] Mon, 4 May 2026 12:34:09 UTC (98 KB) Full-text links: Access Paper: View a PDF of the paper titled Empirical Measures and Strong Laws of Large Numbers in Categorical Probability, by Tobias Fritz and 4 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2025-03 Change to browse by: cs cs.LO math math.CT math.ST stat stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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