  In a quantum Markov chain, the temporal succession of states is modeled by
the repeated action of a "bistochastic quantum operation" on the density matrix
of a quantum system. Based on this conceptual framework, we derive some new
results concerning the evolution of a quantum system, including its long-term
behavior. Among our findings is the fact that the Ces$\grave{a}$ro limit of any
quantum Markov chain always exists and equals the orthogonal projection of the
initial state upon the eigenspace of the unit eigenvalue of the bistochastic
quantum operation. Moreover, if the unit eigenvalue is the only eigenvalue on
the unit circle, then the quantum Markov chain converges in the conventional
sense to the said orthogonal projection. As a corollary, we offer a new
derivation of the classic result describing limiting distributions of unitary
quantum walks on finite graphs \cite{AAKV01}.
