Central Limits via Dilated Categories

2026-03-09

AI Summary

This work addresses the absence of a unified theoretical framework for the Central Limit Theorem (CLT), which has led to fragmented proofs and obscured insights into its intrinsic structure. To resolve this, the authors propose an abstract framework grounded in enriched category theory with expansive seminorms, uniquely integrating categorical methods with expansion structures to systematically characterize and derive diverse CLTs. This approach not only recovers and strengthens classical results—including the standard CLT and the Law of Large Numbers—but also extends them to statistical mechanical contexts, yielding a novel CLT for observables on symplectic manifolds. The framework thus establishes a categorical foundation for probabilistic limit theorems and opens avenues for broad theoretical and applied generalizations.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Causality

Application Category

• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Central Limit Theorem
• unifying framework
• statistical reasoning
• abstract probability
• limit theorems

Innovation

• dilated categories
• central limit theorem
• seminorm-enriched category theory
• symplectic manifolds
• abstract probability