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On the asymptotic uniform distribution of sums reduced mod 1. (English) Zbl 0557.60008
Let $$G_ n(x)$$ be the distribution function of a sum reduced mod 1 of n independent identically distributed random variables. For the exponential convergence of $$G_ n(x)$$ (or its densities $$p_ n(x))$$ to the uniform distribution, necessary and sufficient conditions are given. One such condition is $$\sup_{r\neq 0}| f(r)| <1$$, where the f(r) are the Fourier coefficients of a summand. If this condition is not satisfied, the rate of convergence can be prescribed arbitrarily. The case of the summands having a lattice distribution is considered in a further paper submitted to the same journal.

##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60B10 Convergence of probability measures
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##### References:
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