Central Limits via Dilated Categories

TL;DR

This paper introduces a unifying categorical framework called dilated seminorm-enriched category theory to generalize and abstract the Central Limit Theorem, including new results for symplectic manifolds and observables.

Contribution

It develops a novel categorical approach to central limits, providing a unified theory that encompasses classical and new CLT variants, including applications in statistical mechanics.

Findings

Unified categorical framework for CLT and law of large numbers

Derived a new CLT for symplectic manifolds and observables

Strengthened classical CLT results within the framework

Abstract

The Central Limit Theorem (CLT) establishes that sufficiently large sequences of independent and identically distributed random variables converge in probability to a normal distribution. This makes the CLT a fundamental building block of statistical reasoning and, by extension, in reasoning about computing systems that are based on statistical inference such as probabilistic programing languages, programs with optimisation, and machine learning components. However, there is no general theory of CLT-like results currently, which forces practitioners to redo proofs without having a good handle on the essential ingredients of CLT-type results. In this paper, we introduce dilated seminorm-enriched category theory as a unifying framework for central limits, and we establish an abstract central limit theorem within that framework. We illustrate how a strengthened version of the classical CLT and the law of large numbers can be obtained as instances of our framework. Moreover, we derive from our framework a novel central limit theorem for symplectic manifolds, the CLT for observables, which finds applications in statistical mechanics.