On discrete approximations of stable distributions
  In some fields of applications of stable distributions, especially in economics, it appears, that data have distributions similar to stable in a large region, but do not have such heavy tails. Our aim in this note is to propose several methods of approximation of stable distributions by some discrete distributions, which may have different tail behavior. In a sense the introduced distributions form an alternative to tempered stable distributions that combine Gaussian and stable behavior. 
Lenka Sl\'amov\'a and Lev B. Klebanov
math.PR math.ST stat.TH