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The modulo 1 central limit theorem and Benford’s law for products. (English) Zbl 1148.60008
Let $$B$$ be any base. Then a positive $$x\in \mathbb{R}$$ can be written as $$x=M_B(x)\cdot B^k$$ with $$k\in \mathbb{Z}$$, where the mantissa $$M_B(x)\in[1, B)$$. A sequence of positive integers $$\{a_n\}$$ is said to satisfy Benford’s law base $$B$$ if $\lim_{N\to\infty}{\#\{n\leq N: 1\leq M_B(a_n)\leq s\}\over N}=\log_B s.$ First, the authors find the necessary and sufficient condition in terms of Fourier coefficients of a probability density for the sum of independent continuous random variables modulo $$1$$ to converge to the uniform distribution in $$L^1([0,1])$$, and apply this for the convergence to the Benford law base $$B$$. Namely, let $$X_1,\dots, X_M$$ be independent continuous random variables, $$g_{B, m}$$ denote the density of $$\log_BM_B(| X_m| )$$, and let $$\widehat{g}_{B, m}(n)$$ be the $$n$$th Fourier coefficient of $$g_{B, m}$$. Then, the distribution of the digits of $$X_1 \cdots X_M$$ converges to the Benford’s law base $$B$$ as $$M\to\infty$$ in $$L^1([0, 1])$$ if and only if for each $$n\neq 0$$
$\lim_{M\to\infty}\widehat{g}_{B, 1}(n) \cdots \widehat{g}_{B, M}(n)=0.$ Also, a generalization of a mentioned limit theorem for the sum of independent discrete random variables on $$[0, 1)$$ is given.

##### MSC:
 60F05 Central limit and other weak theorems 60F25 $$L^p$$-limit theorems 11K06 General theory of distribution modulo $$1$$ 42A10 Trigonometric approximation 42A61 Probabilistic methods for one variable harmonic analysis 62E15 Exact distribution theory in statistics
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