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

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.