The main achievement of this paper is the finding and proof of Central Limit Theorem (CLT, see Theorem 12) under the framework of sublinear expectation. Roughly speaking under some reasonable assumption, the random sequence $\{1/\sqrt {n}(X_{1}+\cdots +X_{n})\}_{i=1}^{\infty }$ converges in law to a nonlinear normal distribution, called G-normal distribution, where $\{X_{i}\}_{i=1}^{\infty }$ is an i.i.d. sequence under the sublinear expectation. It’s known that the framework of sublinear expectation provides a important role in situations that the probability measure itself has non-negligible uncertainties. Under such situation, this new CLT plays a similar role as the one of classical CLT. The classical CLT can be also directly obtained from this new CLT, since a linear expectation is a special case of sublinear expectations. A deep regularity estimate of 2nd order fully nonlinear parabolic PDE is applied to the proof of the CLT. This paper is originally exhibited in arXiv.(math.PR/0702358v1).

中文翻译：

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非线性期望下的大数定律和中心极限定理

本文的主要成就是在亚线性期望的框架下发现并证明了中心极限定理（CLT，参见定理12）。粗略地说，在某个合理的假设下，随机序列$ \ {1 / \ sqrt {n}（X_ {1} + \ cdots + X_ {n}）\} _ {i = 1} ^ {\ infty} $收敛于定律称为非线性正态分布，称为G正态分布，其中$ \ {X_ {i} \} _ {i = 1} ^ {\ infty} $是亚线性期望下的iid序列。众所周知，在概率测度本身具有不可忽略的不确定性的情况下，亚线性期望框架发挥了重要作用。在这种情况下，这种新的CLT扮演的角色与经典CLT相似。由于线性期望是次线性期望的特殊情况，因此也可以直接从此新CLT中获得经典CLT。二阶完全非线性抛物线PDE的深度正则性估计被应用于CLT的证明。本文最初在arXiv。（math.PR/0702358v1）中展出。