The Modulo 1 Central Limit Theorem and Benford's Law for Products

Abstract

We derive a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distribution in L^1([0,1]), and discuss generalizations to discrete random variables. A consequence is that if X_1, ..., X_M are independent continuous random variables with densities f_1, ..., f_M, for any base B as M \to \infty for many choices of the densities the distribution of the digits of X_1 * ... * X_M converges to Benford's law base B. The rate of convergence can be quantified in terms of the Fourier coefficients of the densities, and provides an explanation for the prevalence of Benford behavior in many diverse systems.