On the asymptotic uniform distribution of sums reduced mod 1. (English) Zbl 0557.60008

Schatte, Peter

Math. Nachr. 115, 275-281 (1984).

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

Keywords:

sum reduced mod 1; uniform distribution; rate of convergence

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