: Since the early 1960s, there have been a good number of papers related to heavy tail distributions. These papers support the view that the heavy tail property is a stylized fact about financial time series. Stable distributions have infinite variance, a property which is not found in empirical samples where empirical variance does not grow with the size of the sample. As an example of the application of fractional calculus, by extending diffusion equations to fractional order, a connection between stable and tempered stable distribution with fractional calculus can be made.