Chapter 7: Random orthogonal matrices

Abstract

Orthogonal matrices, both fixed and random, play an important role in much of statistics, especially in multivariate analysis. Connections between the orthogonal group 0" and the multivariate normal distribution are explored in James (1954) and in Wijsman (1957), as well as in many texts on multivariate analysis. In this chapter, invariance arguments are used to derive the density of a subblock of a uniformly distributed element of 0". This result is used to describe an upper bound on the rate at which one has convergence (as n ~ co) to the multivariate normal distribution.