Unifying Probabilistic Interpretation of Benford’s Law

We propose a probabilistic interpretation of Benford’s law, which predicts the probability distribution of all digits in everyday-life numbers. Heuri- stically, our point of view consists in considering an everyday-life number as a continuous random variable taking value in an interval [0, A], whose maximum A is itself an everyday-life number. This approach can be linked to the chara- cterization of Benford’s law by scale-invariance, as well as to the convergence of a product of independent random variables to Benford’s law. It also allows to generalize Flehinger’s result about the convergence of iterations of Cesaro- averages to Benford’s law.