  A convergence theorem is obtained for quantum random walks with particles in
an arbitrary normal state. This result unifies and extends previous work on
repeated-interactions models, including that of the author (2010, J. London
Math. Soc. (2) 81, 412-434; 2010, Comm. Math. Phys. 300, 317-329). When the
random-walk generator acts by ampliation and multiplication or conjugation by a
unitary operator, necessary and sufficient conditions are given for the quantum
stochastic cocycle which arises in the limit to be driven by an isometric,
co-isometric or unitary process.
