  We investigate the predictability of extreme events in time series. The focus
of this work is to understand under which circumstances large events are better
predictable than smaller events. Therefore we use a simple prediction algorithm
based on precursory structures which are identified using the maximum
likelihood principle. Using the receiver operator characteristic curve as a
measure for the quality of predictions we find that the dependence on the event
magnitude is closely linked to the probability distribution function of the
underlying stochastic process. We evaluate this dependence on the probability
distribution function analytically and numerically. If we assume that the
optimal precursory structures are used to make the predictions, we find that
large increments are better predictable if the underlying stochastic process
has a Gaussian probability distribution function, whereas larger increments are
harder to predict if the underlying probability distribution function has a
power law tail. In the case of an exponential distribution function we find no
significant dependence on the event magnitude. Furthermore we compare these
results with predictions of increments in correlated data, namely, velocity
increments of a free jet flow. The velocity increments in the free jet flow are
in dependence on the time scale either asymptotically Gaussian or
asymptotically exponential distributed. The numerical results for predictions
within free jet data are in good agreement with the previous analytical
considerations for random numbers.
