Convergence in distribution of the product of random variables from an independent sample on a compact algebraic group

TL;DR

This paper establishes conditions under which the product of independent random variables on a compact algebraic group converges in distribution, highlighting the role of the distribution's support and subgroup structure.

Contribution

It provides an equivalent condition for convergence in distribution of products on compact algebraic groups, linking support properties to the existence of a limit distribution.

Findings

Limit distribution exists and is uniform on the support under certain conditions.

Convergence depends on the distribution's non-membership in non-trivial cosets over algebraic subgroups.

The result characterizes when the product sequence converges in distribution.

Abstract

An equivalent condition for the product of elements of an independent random sample on a compact algebraic group converging in distribution to some random variable as the sample size increases is obtained. Namely, a limit distribution exists and is uniform on the support of the parent distribution if a random variable with such a distribution does not belong with the unit probability to any non-trivial coset over an algebraic subgroup that lies in its normalizer; otherwise, it does not exist.