  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.
