  The Born rule describes the probability of obtaining an outcome when
measuring an observable of a quantum system. As it can only be tested by
measuring many copies of the system under consideration, it cannot hold
strictly for non-replicable systems. For these systems, we give a procedure to
predict the future statistics of measurement outcomes through Repeated
Measurements (RM). We prove that if the statistics of the results acquired via
RM is sufficiently similar to that obtained by the Born rule, the latter can be
used effectively. We apply our framework to a repeatedly measured Unruh-DeWitt
detector interacting with a massless scalar quantum field, which is an example
of a system (detector) interacting with an uncontrollable environment (field)
for which using RM is necessary. Analysing what an observer learns from the RM
outcomes, we find a regime where history-dependent RM probabilities are close
to the Born ones. Consequently, the latter can be used for all practical
purposes. Finally, we study numerically inertial and accelerated detectors
showing that an observer can see the Unruh effect via RM.
