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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
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