Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects

We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler 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 followed by a 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 spatially one-dimensional 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 space-time fractional diffusion equation. Finally, we treat the "time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.