The moment problem for random objects in a category

TL;DR

This paper extends the classical moment problem to random objects in categories, providing criteria for the existence and uniqueness of measures based on moments defined via epimorphisms, with applications in number theory.

Contribution

It introduces a novel moment problem framework for categories satisfying certain finiteness and isomorphism conditions, including finite groups and modules, with explicit formulas and convergence results.

Findings

Necessary and sufficient conditions for measure existence based on moments.

Explicit formulas for measures in terms of moments.

Measures with given moments converge to a unique distribution.

Abstract

The moment problem in probability theory asks for criteria for when there exists a unique measure with a given tuple of moments. We study a variant of this problem for random objects in a category, where a moment is given by the average number of epimorphisms to a fixed object. When the moments do not grow too fast, we give a necessary and sufficient condition for existence of a distribution with those moments, show that a unique such measure exists, give formulas for the measure in terms of the moments, and prove that measures with those limiting moments approach that particular measure. Our result applies to categories satisfying some finiteness conditions and a condition that gives an analog of the second isomorphism theorem, including the categories of finite groups, finite modules, finite rings, as well as many variations of these categories. This work is motivated by the non-abelian Cohen-Lenstra-Martinet program in number theory, which aims to calculate the distribution of random profinite groups arising as Galois groups of maximal unramified extensions of random number fields.