zbMATH — the first resource for mathematics

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.

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60B10 Convergence of probability measures
Full Text: DOI
[1] Stochastic processes. J. Wiley & Sons New York 1953
[2] Probabilistic number theory. vol I. Springer, Berlin 1979 · doi:10.1007/978-1-4612-9989-9
[3] Fainleib, [Russian Text Ignored] 32 pp 859– (1968)
[4] Lehrbuch der Wahrscheinlichkeitsrechnung. Akademie-Verlag Berlin 1968 (Translation from the Russian)
[5] Herrmann, Math. Nachr. 104 pp 49– (1981)
[6] and , Independent and stationary sequences of random variables. Wolters – Nordhoff Groningen 1971 (Translation from the Russian)
[7] Kerstan, I. Math. Nachr. 37 pp 267– (1968)
[8] Kloss, [Russian Text Ignored] 4 pp 255– (1959)
[9] Raimi, Amer. Math. Monthly 83 pp 521– (1976)
[10] Irrationalzahlen. Walter de Gruyter, Berlin 1939
[11] Schatte, Zeitschr. f. Angew. Math. u. Mech. 83 pp 553– (1973)
[12] Schatte, Math. Nachr. 110 pp 243– (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.