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From uniform distributions to Benford’s law. (English) Zbl 1065.60095
The authors consider the mantissa of a real number $$x$$ as the unique real number $$M(x) \in [1,10)$$ such that $$x=M(x)10^k$$, where $$k$$ is an integer. Benford’s law describes the probability distribution of the mantissa, namely, Pr$$(M(x) \in [a,b)) = \log_{10}b - \log_{10}a$$ for any $$a,b$$ such that $$1 \leq a < b \leq 10$$. The characteristic property of Benford’s law is established. The result is applied for the estimation of the rate of convergence in B. J. Flehinger’s theorem [Am. Math. Mon. 73, 1056–1061 (1966; Zbl 0147.17502)].

##### MSC:
 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 11K99 Probabilistic theory: distribution modulo $$1$$; metric theory of algorithms
##### Keywords:
mantissa; probability distribution; Benford’s law
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##### References:
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