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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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