What the study found
The authors develop categorical tools for studying empirical measures and prove abstract versions of the de Finetti theorem, the Glivenko–Cantelli theorem, and the strong law of large numbers from the same framework.
Why the authors say this matters
The study suggests that this framework gives a joint proof of these classical results from first principles. The authors also indicate that it helps analyze limits of empirical measures in categorical probability.
What the researchers tested
The researchers proposed two axioms for morphisms from infinite sequences to single outputs: permutation invariance and empirical adequacy. They worked in quasi-Markov categories, which allow partial morphisms, and constructed empirical sampling morphisms as partially defined Markov kernels on standard Borel spaces.
What worked and what didn't
Given an empirical sampling morphism and additional properties, the authors prove representability and abstract versions of the de Finetti theorem, the Glivenko–Cantelli theorem, and the strong law of large numbers. By instantiating the abstract results, they recover the standard Glivenko–Cantelli theorem and the strong law of large numbers for random variables with finite first moment.
What to keep in mind
Not all sequences have a well-defined empirical measure, so the framework uses partial morphisms. The abstract does not describe other limitations beyond this scope constraint.
Key points
- The paper builds a categorical framework for empirical measures and empirical sampling.
- Two axioms are proposed: permutation invariance and empirical adequacy.
- The authors prove abstract versions of the de Finetti theorem, the Glivenko–Cantelli theorem, and the strong law of large numbers.
- Concrete empirical sampling morphisms are constructed as partially defined Markov kernels on standard Borel spaces.
- The standard Glivenko–Cantelli theorem and strong law of large numbers are recovered for random variables with finite first moment.
Disclosure
- Research title:
- Empirical sampling tools recover classical limit theorems
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