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

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