A discrete parameter quantum process is represented by a sequence of quantum operations, which are completely positive maps that are trace non-increasing. Given a stationary and ergodic sequence of such maps, an ergodic theorem describing convergence to equilibrium for a general class of such processes was recently obtained by Movassagh and Schenker. Under irreducibility conditions, we obtain a law of large numbers that describes the asymptotic behavior of the processes involving the Lyapunov exponent. Furthermore, a central limit-type theorem is obtained under mixing conditions. These results do not require the sequences of quantum operations that describe the quantum process to be trace non-increasing and hence can be applied to a larger class of compositions of random positive maps. In the continuous-time parameter, a quantum process is represented by a double-indexed family of positive map-valued random variables. For a stationary and ergodic family of such maps, we extend the results by Movassagh and Schenker to the continuous case.