Tightness, weak compactness of nonlinear expectations and application to CLT

In this paper we introduce a notion of tightness for a family of nonlinear expectations and show that the tightness can be applied to obtain weak compactness in a framework of nonlinear expectation space. This criterion is very useful for obtaining the weak convergence for a sequence of nonlinear expectations, which is a equivalent to the so-called convergence in distribution, or in law for a sequence of random variables in a nonlinear expectation space. We use the above result to give a new proof to the central limit theorem under a sublinear expectation space. The method can be also applied to prove the convergence of some numerical schemes for degenerate fully nonlinear PDEs.