Positivity 10 (2006), 755–759
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040755-5, published online July 11, 2006
Random Points in the Unit Ball of
Abstract. We show that two limit results from random matrix theory, due to
Marˇcenko–Pastur and Bai–Yin, are also valid for matrices with independent
rows (as opposed to independent entries in the classical theory), when rows
are uniformly distributed on the unit ball of
, under proper normalization.
Mathematics Subject Classiﬁcation (2000). Primary 15A52; 52A21.
spaces, random matrices, random vectors.
Let us start with the following classical results from Random Matrix Theory.
Let Z be a random variable such that
EZ =0 and EZ
Consider an inﬁnite array (Z
) of i.i.d. copies of Z. For each couple (n, N)of
integers, let G
be the N × n random matrix
We consider also the matrix
We may drop subscripts and write simply G and A. The matrix A is sometimes
called a sample covariance matrix. Let (λ
(A)) be the eigenvalues of A, arranged in
decreasing order. We write λ
(A). The spectral
measure of A is the probability measure on R deﬁned as
In other words, µ
(B) is the proportion of eigenvalues of A that fall in a Borel
set B ⊂ R. The following theorems describe the limit behaviour of the spectrum
of such large-dimensional matrices, in both global and local regime.
Research was supported in part by the European Network PHD, FP6 Marie Curie Actions,
MCRN-511953 and was done in part while the author was visiting the University of Athens.