  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.
