  The non-Markovianity of an arbitrary open quantum system is analyzed in
reference to the multi-time statistics given by its monitoring at discrete
times. On the one hand, we exploit the hierarchy of inhomogeneous transfer
tensors, which provides us with relevant information about the role of
correlations between the system and the environment in the dynamics. The
connection between the transfer-tensor hierarchy and the CP-divisibility
property is then investigated, by showing to what extent quantum Markovianity
can be linked to a description of the open-system dynamics by means of the
composition of 1-step transfer tensors only. On the other hand, we introduce
the set of stochastic transfer tensor transformations associated with local
measurements on the open system at different times and conditioned on the
measurement outcomes. The use of the transfer-tensor formalism accounts for
different kinds of memory effects in the multi-time statistics and allows us to
compare them on a similar footing with the memory effects present in
non-monitored non-Markovian dynamics, as we illustrate on a spin-boson case
study.
