Central Limits via Dilated Categories

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.