Monatshefte für Mathematik

, Volume 107, Issue 3, pp 245–256 | Cite as

On measures of uniformly distributed sequences and Benford's law

  • Peter Schatte
Article

Abstract

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 byN−1/2log3N for sufficiently largeN. 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.

Keywords

Distribution Function General Product Uniform Distribution Empirical Distribution Product Measure 

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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Peter Schatte
    • 1
  1. 1.Sektion Mathematik der Bergakademie FreibergFreibergGerman Democratic Republic

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