  By modeling the interaction of an open quantum system with its environment
through a natural generalization of the classical concept of continuous time
random walk, we derive and characterize a class of non-Markovian master
equations whose solution is a completely positive map. The structure of these
master equations is associated with a random renewal process where each event
consist in the application of a superoperator over a density matrix. Strong
non-exponential decay arise by choosing different statistics of the renewal
process. As examples we analyze the stochastic and averaged dynamics of simple
systems that admit an analytical solution. The problem of positivity in quantum
master equations induced by memory effects [S.M. Barnett and S. Stenholm, Phys.
Rev. A 64, 033808 (2001)] is clarified in this context.
