  By modeling the interaction of a system with an environment through a renewal
approach, we demonstrate that completely positive non-Markovian dynamics may
develop some unexplored non-standard statistical properties. The renewal
approach is defined by a set of disruptive events, consisting in the action of
a completely positive superoperator over the system density matrix. The random
time intervals between events are described by an arbitrary waiting-time
distribution. We show that, in contrast to the Markovian case, if one performs
a system-preparation (measurement) at an arbitrary time, the subsequent
evolution of the density matrix evolution is modified. The non-stationary
character refers to the absence of an asymptotic master equation even when the
preparation is performed at arbitrary long times. In spite of this property, we
demonstrate that operator expectation values and operators correlations have
the same dynamical structure, establishing the validity of a non-stationary
quantum regression hypothesis. The non-stationary property of the dynamic is
also analyzed through the response of the system to an external weak
perturbation.
