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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.

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
Full Text: arXiv