Weak Convergence to Stochastic Integrals Driven by $α-$Stable Lévy 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\évy process. Our method is very powerful to obtain the limit behavior of heavy-tailed random variables.