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 $α-$stable Lévy process. Our method is very powerful to obtain the limit behavior of heavy-tailed random variables.

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