On the rate of convergence for an $α$-stable central limit theorem under sublinear expectation
Mingshang Hu, Lianzi Jiang, Gechun Liang
TL;DR
This work analyzes the rate of convergence for the α-stable central limit theorem under sublinear expectation by developing a monotone scheme to approximate a class of fully nonlinear degenerate PIDEs that characterize nonlinear α-stable Lévy processes. The authors construct a sublinear-space framework with a random variable ξ and derive a discrete scheme whose convergence to the PIDE viscosity solution yields an explicit Berry-Esseen-type bound with rate Γ(α,q) = min{1/4, (2-α)/(2α), q/2}. They provide regularity results, a detailed monotone scheme analysis, and separate lower/upper bound arguments to establish sharp convergence rates, including two illustrative examples that show how tail parameters influence the rate. The results bridge nonlinear PDE numerical analysis and probability under sublinear expectation, enabling quantitative CLT approximations under model uncertainty with explicit error bounds.
Abstract
In this paper, we propose a monotone approximation scheme for a class of fully nonlinear degenerate partial integro-differential equations (PIDEs) which characterize the nonlinear $α$-stable Lévy processes under sublinear expectation space with $α\in(1,2)$. We further establish the error bounds for the monotone approximation scheme. This in turn yields an explicit Berry-Esseen bound and convergence rate for the $α$-stable central limit theorem under sublinear expectation.