Title

Finite-state Markov Chains Obey Benford’s Law

Author Info

Babar Kaynar, Vrije University
Arno Berger, University of Alberta
Theodore P. Hill, Georgia Institute of Technology - Main CampusFollow
Ad Ridder, Vrije University

Recommended Citation

Available from Mathematics arXiv, March 5, 2010.

31 pages.

Copyright © 2010 by Babar Kaynar, Arno Berger, Theodore P. Hill and Ad Ridder. The definitive version is available at http://arxiv.org/abs/1003.0562.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

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 (Pⁿ - P*) and (Pⁿ⁺¹-Pⁿ) 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.

Disciplines

Mathematics