On the Kolmogorov Superposition Theorem and Regular Means

TL;DR

This paper reveals a deep connection between Kolmogorov's superposition theorem and the concept of regular means, highlighting their implications for statistical theory, neural models, and stability properties.

Contribution

It demonstrates that regular means can be derived from the Kolmogorov superposition theorem, linking two foundational results and expanding their significance in statistics.

Findings

Regular means encompass arithmetic, geometric, and harmonic means.

A stability property of regular means under small perturbations is established.

Insights into a universal central limit theorem for regular means are provided.

Abstract

While Kolmogorov's probability axioms are widely recognized, it is less well known that in an often-overlooked 1930 note, Kolmogorov proposed an axiomatic framework for a unifying concept of the mean -- referred to as regular means. This framework yields a well-defined functional form encompassing the arithmetic, geometric, and harmonic means, among others. In this article, we uncover an elegant connection between two key results of Kolmogorov by showing that the class of regular means can be derived directly from the Kolmogorov superposition theorem. This connection is conceptually appealing and illustrates that the superposition theorem deserves wider recognition in Statistics -- not only because of its link to regular means as shown here, but also due to its influence on the development of neural models and its potential connections with other statistical frameworks. In addition, we establish a stability property of regular means, showing that they vary smoothly under small perturbations of the generator. Finally, we provide insights into a recent universal central limit theorem that applies to the broad class of regular means.