One of the ways to generalize the Poisson process is to allow inter-arrival times to follow any probability distribution defined on a non-negative, real line. Such extension is called a renewal process. The inter-arrival time's independence and identical distribution are preserved.
In this chapter, we will go through the process with inter-arrival times drawn from the Levy distribution that has the following probability density function:
Here is a scale parameter and is the location parameter. This is a special case of an inverse-gamma distribution and it is from stable distribution family that is quite often employed to describe fat-tailed data.
Levy distribution is defined on . But trades can be placed as near as possible, therefore we assume that the location parameter is zero.