  The prediction and control of rare events is an important task in disciplines
that range from physics and biology, to economics and social science. The Big
Jump principle deals with a peculiar aspect of the mechanism that drives rare
events. According to the principle, in heavy-tailed processes a rare huge
fluctuation is caused by a single event and not by the usual coherent
accumulation of small deviations. We consider generalized L\'evy walks, a class
of stochastic processes with power law distributed step durations, which model
complex microscopic dynamics in the single stretch. We derive the bulk of the
probability distribution and using the big jump principle, the exact form of
the tails that describes rare events. We show that the tails of the
distribution present non-universal and non-analytic behaviors, which depend
crucially on the dynamics of the single step. The big jump estimate also
provides a physical explanation of the processes driving the rare events,
opening new possibilities for their correct prediction.
