Weak Convergence to Stochastic Integrals Driven by $\alpha-$Stable   L\'evy Processes
  We use the martingale convergence method to get the weak convergence theorem on general functionals of partial sums of independent heavy-tailed random variables. The limiting process is the stochastic integral driven by $\alpha-$stable L\'evy process. Our method is very powerful to obtain the limit behavior of heavy-tailed random variables. 
Zhengyan Lin, Hanchao Wang
math.ST stat.TH