Asymptotic Distribution of Projections of High-Dimensional Random Vectors

• Activity Number: 585
• Type: Contributed
• Date/Time: Wednesday, August 4, 2010 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Armine Bagyan*+ and Arkady Tempelman and Bing Li
• Companies: Penn State and Penn State and Penn State
• Keywords: Projections onto random directions ; Diaconis-Freedman theorem ; limit theorems ; dimension reduction

Abstract

In this paper we study the asymptotic distribution of random projections of high-dimensional random vectors. We extend the central limit theorem for the projections of high-dimensional non-random vectors by Diaconis and Freedman (1984) to the case of random vectors, and show that, under certain conditions, the conditional distribution functions of the projections, given the projection direction, converge in probability to a normal distribution. We also consider the inner product of a p-dimensional vector X and a standard normal vector xi. We prove the L^2-convergence of the conditional density function of xi^T X / p^(1/2) given xi to a normal density function when p increases, and the uniform L^2-convergence under some stricter conditions. In addition, we estimate the rate of the uniform L^2-convergence and consider an example when X is uniformly distributed on a sphere.