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On uniform distribution in probability for sequences of products of independent random elements in a compact metric groupKeywords: Weyl's criterion , uniform distribution modulo $1$ , uniform distribution almost surely , asymptotic uniformity in distribution $(mod 1)$ , uniform distribution in probability , random elements in compact metric groups Abstract: We consider the concept of uniform distribution in probability forsequences of random elements in a compact metric group. We prove acriterion for uniform distribution in probability for sequences ofproducts of independent random elements. As a corollary, we obtaina criterion for uniform distribution modulo $1$ in probability forsequences of sums of independent random variables which generalizesthe classical Weyl's criterion.
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