On measures of uniformly distributed sequences and Benford's law
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- Schatte, P. Monatshefte für Mathematik (1989) 107: 245. doi:10.1007/BF01300347
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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.