Finite-State Markov Chains obey Benford's Law

This discussion paper resulted in a publication in the Siam Journal on Matrix Analysis and Applications (2011). Volume 32, issue 3, pages 665-684. A sequence of real numbers ( x n ) is Benford if the significands, i.e. the fractionparts in the floating-point representation of ( x n ), are distributed logarithmically.Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain withprobability transition matrix P and limiting matrix P* is Benford if every componentof both sequences of matrices ( P n − P* ) and ( P n+1 − P n ) is Benford oreventually zero. Using recent tools that established Benford behavior both forNewton's method and for finite-dimensional linear maps, via the classical theoriesof uniform distribution modulo 1 and Perron-Frobenius, this paper derives asimple sufficient condition ('nonresonance') guaranteeing that P , or the Markovchain associated with it, is Benford. This result in turn is used to show thatalmost all Markov chains are Benford, in the sense that if the transition probabilitiesare chosen independently and continuously, then the resulting Markovchain is Benford with probability one. Concrete examples illustrate the variouscases that arise, and the theory is complemented with several simulations andpotential applications.