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We propose a monotone approximation scheme for a class of fully nonlinear degenerate partial integro-differential equations which characterize nonlinear 
$\alpha$-stable Lévy processes under a sublinear expectation space with 
$\alpha\in(1,2)$. 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 
$\alpha$-stable central limit theorem under sublinear expectation.
