I think that the definition of fractional Brownian Motion is widely [known](http://en.wikipedia.org/wiki/Fractional_Brownian_motion) (for example as a Gaussian Process with particular variance covariance stucture parametrized by the so-called Hurst index). 


Heuristically, you can think of those processes as Gaussian processes with long (or short) memory depending on the value of their Hurst Index, and for Hurst index equal to 1/2 you get classical Brownian Motion (which has no memory).


I was wondering what would be the definition for "fractional Poisson Processes" and what stylised facts about fractional Brownian Motion one should consider in extending the definition to Poisson process. 


If any reference exists about this, this is just fine for me.


I have no other motivation than curiosity on this topic. 



What would be a fractional Poisson Process like
A standard Poisson process is a renewal process with exponential distributed waiting times. Fractional Poisson process (FPP) is also a renewal process with Mittag-Leffler waiting times. Note that Mittag-Leffler distribution is a heavy tailed generalization of exponential distribution. Further, let N(t) be a standard Poisson process and $E\_{\alpha}(t) = \inf\{s \geq 0: S\_{\alpha}(s)>t\}$ be the first-exit time of a stable subordinator $S\_{\alpha}(t)$ then the time-changed process $N^\*(t) = N(E\_{\alpha}(t))$ is also a characterization of FPP. 


