  We study rare events in the extreme value statistics of stochastic symmetric
jump processes with power tails in the distributions of the jumps, using the
big jump principle. The principle states that in the presence of stochastic
processes with power tails statistics, if at a certain time a physical quantity
takes on a value much larger than its typical value, this large fluctuation is
realised through a single macroscopic jump that exceeds the typical scale of
the process by several orders of magnitude. In particular, our estimation
focuses on the asymptotic behaviour of the tail of the probability distribution
of maxima, a fundamental quantity in a wide class of stochastic models used in
chemistry to estimate reaction thresholds, in climatology for earthquake risk
assessment, in finance for portfolio management, and in ecology for the
collective behaviour of species. We determine the analytical form of the
probability distribution of rare events in the extreme value statistics of
three L\'evy processes; L\'evy flights, L\'evy walks and the L\'evy-Lorentz
gas. For the L\'evy flights, we re-obtain through the big jump approach recent
analytical results, extending their validity. For the L\'evy-Lorentz gas we
show that the topology of the disordered lattice along which the walker moves
induces memory effects in its dynamics, which influences the extreme value
statistics. Our results are confirmed by extensive numerical simulations.
