A central limit theorem for convex sets
Abstract
Suppose X is a random vector, that is distributed uniformly in some ndimensional convex set. It was conjectured that when the dimension n is very large, there exists a nonzero vector u, such that the distribution of the real random variable <X,u> is close to the gaussian distribution. A wellunderstood situation, is when X is distributed uniformly over the ndimensional 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.
 Publication:

Inventiones Mathematicae
 Pub Date:
 January 2007
 DOI:
 10.1007/s0022200600288
 arXiv:
 arXiv:math/0605014
 Bibcode:
 2007InMat.168...91K
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Probability
 EPrint:
 41 pages