  Memory effects can be studied through a conditional past-future correlation,
which measures departure with respect to a conditional past-future independence
valid in a memoryless Markovian regime. In a quantum regime this property leads
to an operational definition of quantum non-Markovianity based on three
consecutive system measurement processes and postselection [Budini, Phys. Rev.
Lett. 121, 240401 (2018)]. Here, we study the conditional past-future
correlation for a qubit system coupled to different dephasing environments.
Exact solutions are obtained for a quantum spin bath as well as for classically
fluctuating random Hamiltonian models. The developing of memory effects and
departures from Born-Markov or white-noise approximations are related to a
measurement back action that changes the system dynamics between consecutive
measurements. It is shown that this effect may develop even when the former
system evolution is given by a time-independent Lindblad equation. This unusual
non-Markovian case arises when the characteristic parameters of the dynamics
become Lorentzian random distributed variables.
