  We discuss some applications of the Mittag-Leffler function and related
probability distributions in the theory of renewal processes and continuous
time random walks. In particular we show the asymptotic (long time) equivalence
of a generic power law waiting time to the Mittag-Leffler waiting time
distribution via rescaling and respeeding the clock of time. By a second
respeeding (by rescaling the spatial variable) we obtain the diffusion limit of
the continuous time random walk under power law regimes in time and in space.
Finally, we exhibit the time-fractional drift process as a diffusion limit of
the fractional Poisson process and as a subordinator for space-time fractional
diffusion.
