Suppose X is a random vector, that is distributed uniformly in some
n-dimensional convex set. It was conjectured that when the dimension n is very
large, there exists a non-zero vector u, such that the distribution of the real
random variable <X,u> is close to the gaussian distribution. A well-understood
situation, is when X is distributed uniformly over the n-dimensional cube. In
this case, <X,u> is approximately gaussian for, say, the vector u = (1,...,1) /
sqrt(n), as follows from the classical central limit theorem.
We prove the conjecture for a general convex set. Moreover, when the
expectation of X is zero, and the covariance of X is the identity matrix, we
show that for 'most' unit vectors u, the random variable <X,u> is distributed
approximately according to the gaussian law. We argue that convexity - and
perhaps geometry in general - may replace the role of independence in certain
aspects of the phenomenon represented by the central limit theorem.

该研究证明了当随机向量服从凸集分布时，当维度 n 很大时，存在一个非零向量 u，使得实随机变量 <X，u> 的分布接近于高斯分布，并且证明了当 X 的期望为零，协方差为单位矩阵时，对于 “大多数” 单位向量 u，随机变量 <X，u> 近似地服从高斯分布。

凸集的中心极限定理

A Central Limit Theorem for Convex Sets

We present estimators for entropy and other functions of a discrete
probability distribution when the data is a finite sample drawn from that
probability distribution. In particular, for the case when the probability
distribution is a joint distribution, we present finite sample estimators for
the mutual information, covariance, and chi-squared functions of that
probability distribution.

本文介绍了在数据是从离散概率分布抽取的有限样本时，对熵和其他函数的估计器，尤其是当概率分布为联合分布时，我们提出了对该概率分布的互信息、协方差和卡方函数的有限样本估计器。