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On the almost sure convergence of floating-point mantissas and Benford’s law. (English) Zbl 0645.60038
Let \(Y_ 1\), \(Y_ 2\),... be a sequence of random variables and let \(M_ n\) be the floating-point mantissa of \(Y_ n\). Further let \(1_{[1,x)}(\cdot)\) denote the indicator function of the interval [1,x). If \(Y_ n/n\to Z\) a.s., where \(Z\neq 0\) is a further random variable, then the sequence \(1_{[1,x)}(M_ n)\) converges a.s. to log x in the sense of \({\mathcal H}\infty\)-means and logarithmic means, respectively. The speed of convergence in this relations is estimated. As a conclusion, a further argument for Benford’s law is provided.

MSC:
60F15 Strong limit theorems
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