Describing the Behavior of Eigenvectors of Random Matrices Using Sequences of Measures on Orthogonal Groups

A conjecture has previously been made on the chaotic behavior of the eigenvectors of a class of n-dimensional random matrices, where n is very large [J. Silverstein, SIAM J. Appl. Math., 37 (1979), pp. 235–245]. Evidence supporting the conjecture has been given in the form of two limit theorems, as n → ∞, relating the random matrices to matrices formed from the Haar measure, hₙ, on the orthogonal group Oₙ. The present paper considers a reformulation of the conjecture in terms of sequences of the form μ ₙ, where for each n, μ ₙ is a Borel probability measure on Oₙ. A characterization of μ ₙ being “close” to hₙ for n large is developed. It is suggested that before a definition of what it means for \ μ ₙ \ to be asymptotic Haar is decided, properties \ hₙ \ possess should first be proposed as possible necessary conditions. The limit theorems are converted into properties on \ μ ₙ \. It is shown (Theorem 1) that one property is a consequence of the other. Another property is proposed resulting in the construction of measures on $D = D[0,1]$ which converge weakly. It is shown (Theorem 2) that under this necessary condition for asymptotic Haar, not only is the conjecture in general not true, but that the behavior of the eigenvectors of large dimensional sample covariance matrices deviates significantly from being Haar distributed when the i.i.d. standardized components making up the matrix differ in the fourth moment from 3.