Empirical Measures and Strong Laws of Large Numbers in Categorical Probability

Antonio Lorenzin; Areeb Shah Mohammed; Paolo Perrone; Tobias Fritz; Tom\'a\v{s} Gonda

arxiv: 2503.21576 · v4 · submitted 2025-03-27 · 🧮 math.PR · cs.LO· math.CT· math.ST· stat.TH

Empirical Measures and Strong Laws of Large Numbers in Categorical Probability

Tobias Fritz , Tom\'a\v{s} Gonda , Antonio Lorenzin , Paolo Perrone , Areeb Shah Mohammed This is my paper

Pith reviewed 2026-05-22 22:24 UTC · model grok-4.3

classification 🧮 math.PR cs.LOmath.CTmath.STstat.TH

keywords categorical probabilityempirical measuresGlivenko-Cantelli theoremstrong law of large numbersde Finetti theoremquasi-Markov categoriespermutation invarianceempirical adequacy

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The pith

Morphisms satisfying permutation invariance and empirical adequacy enable abstract proofs of de Finetti, Glivenko-Cantelli, and strong law theorems in categorical probability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops tools in categorical probability to study limits of empirical measures. It defines an empirical sampling morphism from infinite sequences to a single output via two axioms: permutation invariance and empirical adequacy. These live in quasi-Markov categories to allow partial maps where empirical measures may not exist. Given such a morphism plus a few other properties, the work proves representability results along with abstract versions of the de Finetti theorem, the Glivenko-Cantelli theorem, and the strong law of large numbers. Concrete realizations as partially defined Markov kernels on standard Borel spaces recover the classical statements for random variables with finite first moment, yielding a joint proof of the three theorems from first principles.

Core claim

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.

What carries the argument

The empirical sampling morphism, a partially defined map of type X to the power of natural numbers to X in a quasi-Markov category that obeys permutation invariance and empirical adequacy.

If this is right

Abstract de Finetti theorem holds under the stated conditions on the morphism.
Abstract Glivenko-Cantelli theorem follows directly from the axioms.
Abstract strong law of large numbers holds for the same morphisms.
The three theorems receive a joint proof from the same set of first principles.
Concrete constructions on Borel spaces match the classical statements for finite first moment.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same axioms might organize proofs of other limit theorems once suitable morphisms are identified in the same categories.
Quasi-Markov categories could let the framework handle empirical measures that are only partially defined on non-standard spaces.
The representability results may connect empirical sampling to other notions of conditional independence already studied in categorical probability.

Load-bearing premise

A morphism of type X to the power of natural numbers to X must satisfy permutation invariance and empirical adequacy in order to count as taking an infinite sequence as input and producing a sample from its empirical measure.

What would settle it

A morphism on standard Borel spaces that meets both permutation invariance and empirical adequacy yet fails to recover the classical Glivenko-Cantelli or strong law statements when the abstract theorems are applied.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Two axioms on empirical sampling morphisms in quasi-Markov categories yield abstract de Finetti, Glivenko-Cantelli and SLLN that recover the classical theorems via concrete kernels on Borel spaces.

read the letter

The main takeaway is that two axioms—permutation invariance and empirical adequacy—on a partial morphism X^ℕ → X suffice to prove representability plus abstract versions of de Finetti, Glivenko-Cantelli, and the strong law, which then specialize to the usual statements on standard Borel spaces, including the SLLN for finite first moments. The paper also supplies explicit partially defined kernels that meet the axioms, so the abstract results apply directly and give a joint derivation of the three theorems from the category structure and the new axioms. That joint aspect and the concrete constructions are the parts that work cleanly. The partiality mechanism in quasi-Markov categories handles sequences without well-defined empirical measures without extra fuss, and the stress-test architecture looks internally consistent with no evident circularity. The axioms are tailored to the sampling interpretation, which makes the theorems follow by design rather than from deeper properties; that is standard for this style of work and not a flaw here. The paper is aimed at people already working in categorical probability or measure-theoretic probability with an interest in unified or abstract treatments. A reader in that niche gets value from the framework and the recovery of classical results. It deserves peer review because the concrete instantiations make the claims checkable and the contribution is a genuine addition to the subfield.

Referee Report

0 major / 3 minor

Summary. The paper develops a framework in quasi-Markov categories for empirical measures by introducing empirical sampling morphisms of type X^ℕ → X that satisfy permutation invariance and empirical adequacy. It proves representability of such morphisms and derives abstract versions of de Finetti's theorem, the Glivenko-Cantelli theorem, and the strong law of large numbers. Concrete partially defined Markov kernels on standard Borel spaces are constructed to instantiate the abstract results and recover the classical theorems, yielding a joint proof from first principles.

Significance. If the derivations hold, the work supplies a unified categorical treatment of these limit theorems that handles partiality naturally. The explicit constructions of empirical sampling morphisms as kernels and their direct instantiation to recover the standard Glivenko-Cantelli and SLLN statements (for finite first moment) constitute a clear strength, as does the joint derivation with de Finetti from the two axioms. This advances categorical probability by providing falsifiable abstract statements that specialize correctly to classical probability.

minor comments (3)

The introduction would benefit from a short paragraph situating the two new axioms against prior categorical treatments of exchangeability or laws of large numbers (e.g., references to existing work on Markov categories and de Finetti).
[§2] Notation for partial morphisms and the precise domain of the empirical sampling morphism (sequences without well-defined empirical measures) could be illustrated with a small diagram or example in §2 to improve readability.
[abstract SLLN statement] In the statement of the abstract SLLN, clarify whether the finite-first-moment hypothesis is inherited from the concrete kernels or is an additional assumption on the category; this affects how directly the classical result is recovered.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. We are pleased that the significance of the unified categorical framework, the explicit constructions, and the joint derivation of de Finetti, Glivenko-Cantelli, and the SLLN were recognized. The recommendation for minor revision is noted; since no specific major comments were raised, we will address any minor editorial or presentational points in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from axioms

full rationale

The paper defines empirical sampling morphisms via two new axioms (permutation invariance and empirical adequacy) in quasi-Markov categories, then derives representability plus abstract de Finetti, Glivenko-Cantelli and SLLN statements directly from those axioms plus a short list of additional properties. Concrete kernels on Borel spaces are exhibited separately to instantiate the abstract results. No step reduces a claimed prediction or theorem to a fitted parameter, self-referential definition, or unverified self-citation chain; the axioms are chosen precisely to capture the target behavior and the proofs proceed from them without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on the two newly proposed axioms that define empirical sampling morphisms and on the background structure of quasi-Markov categories that permit partial maps.

axioms (2)

ad hoc to paper permutation invariance
A morphism X^N to X must be invariant under finite permutations of the input sequence.
ad hoc to paper empirical adequacy
The morphism must output a sample distributed according to the empirical measure of the input sequence.

invented entities (1)

empirical sampling morphism no independent evidence
purpose: A partial morphism that samples from the empirical measure of an infinite sequence
Defined exactly by the two axioms inside quasi-Markov categories.

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