Finite-state Markov Chains obey Benford's Law

TL;DR

This paper establishes a simple nonresonant condition under which finite-state Markov chains obey Benford's Law, showing that almost all such chains are Benford when transition probabilities are chosen randomly.

Contribution

It introduces a new sufficient condition for Markov chains to be Benford and proves that almost all randomly chosen chains satisfy this condition, extending understanding of Benford behavior in stochastic processes.

Findings

Almost all Markov chains with randomly chosen transition probabilities are Benford.

The nonresonant condition guarantees Benford behavior in finite-state Markov chains.

The paper provides concrete examples, simulations, and potential applications of the theory.

Abstract

A sequence of real numbers (x_n) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x_n) are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (P^n - P*) and (P^{n+1}-P^n) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton's method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition (nonresonant) guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probabilities are chosen independently and continuously, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with several simulations and potential applications.