Statistical Inference for inter-arrival times of extreme events in bursty time series

Abstract: 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."