Recent Results about the Largest Eigenvalue of Random Covariance Matrices and Statistical Application∗

Sample covariance matrices are a fundamental tool of multivariate statistics. After data collection, we get an n×p data matrix X. We will call n the number of observations and p the number of predictors. The rows of X are assumed to be realizations of a random variable whose covariance structure is Σ p. For practical applications, one often wishes to estimate Σp in order to understand the dependence structure of the predictors, do various tests etc. Here the sample covariance matrix XX/n plays a key role. (Of course, in practice, it is often computed as (X− X̄)∗(X− X̄)/(n−1), where X̄ stands for the column-wise mean of the matrix X, but for the sake of this note we will assume that the entries are centered.) One most important statistical application in which eigenvalues of the covariance matrix play a key role is Principal Component Analysis (PCA). It is a linear dimensionality reduction procedure, which can also be thought of as a model selection technique. The idea is as follows. We are interested in recovering as much of the total variance in the data as possible while reducing the dimensionality of the problem from p to k. In other words, we