  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.
