The asymptotic universality of the Mittag-Leffler waiting time law in continuous time random walks Rudolf GORENFLO 1 and Francesco MAINARDI 2 1 Department of Mathematics and Informatics, Free University of Berlin, Arnimallee 3, D-14195 Berlin, Germany E-mail address: gorenflo@mi.fu-berlin.de 2 Department of Physics, University of Bologna, and INFN, Via Irnerio 46, I-40126 Bologna, Italy E-mail address: mainardi@bo.infn.it Invited lecture by R. Gorenflo at the 373. WE-Heraeus-Seminar on Anomalous Transport: Experimental Results and Theoretical Challenges, Physikzentrum Bad-Honnef (Germany), 12-16 July 2006. For comments/suggestions please contact: gorenflo@mi.fu-berlin.de Revised version: November 24, 2006 Abstract We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler distribution, the waiting time distribution characteristic for a time-fractional continuous time random walk. This asymptotic equivalence is effected by a combination of “rescaling” time and “respeeding” the relevant renewal process and subsequent passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. As far as we know such procedure has been first applied in the Sixties of the past century by Gnedenko and Kovalenko in their theory of “thinning” a renewal process. Turning our attention to continuous time random walks with a generic power law jump distribution, ”rescaling” space can be interpreted as a second kind of “respeeding” which then, again under a proper relation between the relevant parameters leads in the limit to the time-space fractional diffusion equation. Finally, we treat the ‘time-fractional drift” process as properly scaled limit of the counting number of a Mittag-Leffler renewal process. 1