Giuliano, Rita; Janvresse, Élise A unifying probabilistic interpretation of Benford’s law. (English) Zbl 1274.60229 Unif. Distrib. Theory 5, No. 2, 169-182 (2010). The sequence \(x_n\), \(n=1,2,\dots \), of positive numbers satisfies Benford’s law (generalized) in base \(b\), if for every integer \(K\) the relative density of terms \(x_n\), such that the base \(b\) representation of \(x_n\) starts with digits \(K\) (the first zero digits are omitted) is equal to \(\log _b\bigl (1+(1/K)\bigr)\). A basic result is that \(x_n\) satisfies Benford’s law if and only if the sequence \(\log _b x_n\bmod 1\) is uniformly distributed in \([0,1]\).In this paper, the authors present a probabilistic interpretation of Benford’s law. Let \(M_b(x)\) be the mantissa of \(x>0\) defined by \(x=M_b(x)\times b^{n(x)}\) such that \(1\leq M_b(x)<b\) holds, where \(n(x)\) is a uniquely determined integer. A positive random variable \(X\) satisfies Benford’s law if \({\operatorname {Prob}}\bigl (M_b(X)<t\bigr)=\log _bt\). Firstly, the authors prove that if \(X=AY\), where \(Y\) is a continuous random variable and \(A\) is a positive random variable independent of \(Y\), and \(M(A)\) and \(M(X)\) follow the same probability distribution, then this distribution is Benford’s law. Secondly, they consider a Markov chain, defined as the mantissa of a product of independent random variables. Under some conditions, the mantissa of such a product converges to Benford’s law. The authors give a construction of the chain with exponential convergence (cf. [D. Jang et al., J. Algebra Number Theory, Adv. Appl. 1, No. 1, 37–60 (2009; Zbl 1180.11026)]). Reviewer: Oto Strauch (Bratislava) Cited in 1 Document MSC: 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. Keywords:first digit problem; mantissa; exponential speed of convergence; averaging method Citations:Zbl 1180.11026 × Cite Format Result Cite Review PDF