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On the inequalities of Erdős-Turán and Berry-Esseen. I. (English) Zbl 0669.10072

This note contains the announcement of various generalizations and improvements of the Erdős-Turán and Berry-Esseen inequalities for distances of distribution functions. The new ingredient is the use of the modulus of nonmonotonicity instead of the modulus of continuity. Several earlier inequalities such as those of A. S. Fajnlejb [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 859-879 (1968; Zbl 0164.055)] and of the reviewer and W. Philipp [Duke Math. J. 40, 633-649 (1973; Zbl 0273.10043)] are subsumed in the present results.
Reviewer: H.Niederreiter

MSC:

11K06 General theory of distribution modulo \(1\)
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References:

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[2] A. S. Fainleib: A generalization of Esseen’s inequality and its application in probabilistic number theory. Izv. Akad. Nauk SSSR, Ser. Mat., 32, 859-879 (1968) (in Russian). · Zbl 0164.05504
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