  In many complex systems studied in statistical physics, inter-arrival times
between events such as solar flares, trades and neuron voltages follow a
heavy-tailed distribution. The set of event times is fractal-like, being dense
in some time windows and empty in others, a phenomenon which has been dubbed
"bursty". A new model for the inter-exceedance times of events above high
thresholds is proposed. For high thresholds and infinite-mean waiting times, it
is shown that the times between threshold crossings are Mittag-Leffler
distributed, and thus form a "fractional Poisson Process" which generalizes the
standard Poisson Process of threshold exceedances. Graphical means of
estimating model parameters and assessing model fit are provided. The inference
method is applied to an empirical bursty time series, and it is shown how the
memory of the Mittag-Leffler distribution affects prediction of the time until
the next extreme event."
