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

The goal of this expository paper is to describe conditions which guarantee a
central limit theorem for functionals of general state space Markov chains.
This is done with a view towards Markov chain Monte Carlo settings and hence
the focus is on the connections between drift and mixing conditions and their
implications. In particular, we consider three commonly cited central limit
theorems and discuss their relationship to classical results for mixing
processes. Several motivating examples are given which range from toy
one-dimensional settings to complicated settings encountered in Markov chain
Monte Carlo.

本文旨在描述通用状态空间马尔可夫链的泛函的中心极限定理的保证条件，重点关注漂移和混合条件及其含义，包括三个常见引用的中心极限定理和与混合过程的经典结果的关系。通过多个实例阐明了本著作的研究方法和成果。