Abstract: Heavy-tailed inter-arrival times are a signature of ``bursty’’ dynamics, and have been observed in financial time series, earthquakes, solar flares and neuron spike trains. We propose to model extremes of such time series via a ``Max-Renewal process’’ (aka ``Continuous Time Random Maxima process’’). Due to geometric sum-stability, the inter-arrival times between extremes are attracted to a Mittag-Leffler distribution: As the threshold height increases, the Mittag-Leffler shape parameter stays constant, while the scale parameter grows like a power-law. Although the renewal assumption is debatable, this theoretical result is observed for many datasets. We discuss approaches to fit model parameters and assess uncertainty due to threshold selection.