  In this paper, we answer the question of equivalence, or singularity, of two
given quasi-stationary Markov measures on one-sided infinite words, and the
corresponding question of equivalence of associated Cuntz algebra $\O_N$
representations. We do this by associating certain monic representations of
$\O_N$ to quasi-stationary Markov measures, and then proving that equivalence
for pairs of measures is decided by unitary equivalence of the corresponding
pair of representations.
