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

TL;DR

This paper explores the Newcomb-Benford law's scale invariance, introduces a Markov process model that converges to this law, and discusses its derivation, applications, and significance in understanding digit distributions.

Contribution

It presents a simple Markov process model inspired by scale invariance that irreversibly converges to the Newcomb-Benford law, linking it to thermodynamic concepts.

Findings

The Markov process converges to the Benford distribution.

The law is derived from scale invariance principles.

Applications of the law are discussed.

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.