Weak Convergence to Stochastic Integrals Driven by $\alpha-$Stable L\'evy Processes

TL;DR

This paper establishes a weak convergence theorem for functionals of heavy-tailed random variables, showing they converge to stochastic integrals driven by alpha-stable Lévy processes, using martingale methods.

Contribution

It introduces a powerful martingale convergence approach to analyze the limit behavior of heavy-tailed random variables converging to stable Lévy-driven stochastic integrals.

Findings

Proves weak convergence to alpha-stable Lévy-driven stochastic integrals.

Demonstrates the effectiveness of martingale methods for heavy-tailed variables.

Provides a general framework for analyzing heavy-tailed sum functionals.

Abstract

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.