The Newcomb-Benford law: Scale invariance and a simple Markov process based on it

Abstract

The Newcomb-Benford law, also known as the first-digit law, gives the probability distribution associated with the first digit of a dataset, so that, for example, the first significant digit has a probability of $30.1$ % of being $1$ and $4.58$ % of being $9$. This law can be extended to the second and next significant digits. This article presents an introduction to the discovery of the law, its derivation from the scale invariance property, as well as some applications and examples, are presented. Additionally, a simple model of a Markov process inspired by scale invariance is proposed. Within this model, it is proved that the probability distribution irreversibly converges to the Newcomb-Benford law, in analogy to the irreversible evolution toward equilibrium of physical systems in thermodynamics and statistical mechanics.