An $\alpha$-stable limit theorem under sublinear expectation

TL;DR

This paper establishes a generalized central limit theorem for alpha-stable random variables under sublinear expectation, using PDE techniques, extending classical results to a nonlinear expectation framework.

Contribution

It introduces a novel proof approach for the alpha-stable limit theorem under sublinear expectation, avoiding traditional characteristic function methods.

Findings

Generalized CLT for alpha-stable variables under sublinear expectation

Interior regularity estimates for PIDEs are key to the proof

Classical CLT recovered under additional conditions

Abstract

For $\alpha\in (1,2)$, we present a generalized central limit theorem for $\alpha$-stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, provided a mild but natural additional condition holds. Our approach contrasts with previous arguments for the result in the linear setting which have typically relied upon tools that are non-existent in the sublinear framework, for example, characteristic functions.