  We investigate the process of quantum measurements on scattered probes.
Before scattering, the probes are independent, but they become entangled
afterwards, due to the interaction with the scatterer. The collection of
measurement results (the history) is a stochastic process of dependent random
variables. We link the asymptotic properties of this process to spectral
characteristics of the dynamics. We show that the process has decaying time
correlations and that a zero-one law holds. We deduce that if the incoming
probes are not sharply localized with respect to the spectrum of the
measurement operator, then the process does not converge. Nevertheless, the
scattering modifies the measurement outcome frequencies, which are shown to be
the average of the measurement projection operator, evolved for one interaction
period, in an asymptotic state. We illustrate the results on a truncated
Jaynes-Cummings model.
