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On measures of uniformly distributed sequences and Benford’s law. (English) Zbl 0678.10036
The metric theory of uniform distribution of sequences is complemented by considering product measures with not necessarily identical factors. A necessary and sufficient condition is given under which a general product measure assigns the value one to the set of uniformly distributed sequences. For a stationary random product measure, almost all sequences are uniformly distributed with probability one. The discrepancy is estimated by \(N^{-1/2}\log^ 3N\) for sufficiently large N. Thus the metric predominance of uniformly distributed sequences is stated, and a further explanation for Benford’s law is provided. The results can also be interpreted as estimates of the empirical distribution function for non-identical distributed samples.
Reviewer: P.Schatte
MSC:
11K06 General theory of distribution modulo \(1\)
60F15 Strong limit theorems
62G30 Order statistics; empirical distribution functions
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