## The Annals of Probability

### Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices

#### Abstract

Let $B _n = (1/N) T_n^{1/2} X _n X _n^*T_n^{1/2}$ where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. It was shown in an earlier paper by the authors that, under certain conditions on the eigenvalues of $T_n$, with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution (known to exist) for all large $n$. For these $n$ the interval corresponds to one that separates the eigenvalues of $T_n$. The aim of the present paper is to prove exact separation of eigenvalues; that is, with probability 1, the number of eigenvalues of $B_n$ and $T_n$ lying on one side of their respective intervals are identical for all large $n$.

#### Article information

Source
Ann. Probab., Volume 27, Number 3 (1999), 1536-1555.

Dates
First available in Project Euclid: 29 May 2002

https://projecteuclid.org/euclid.aop/1022677458

Digital Object Identifier
doi:10.1214/aop/1022677458

Mathematical Reviews number (MathSciNet)
MR1733159

Zentralblatt MATH identifier
0964.60041

Subjects
Primary: 15A52 60F15: Strong theorems
Secondary: 62H99: None of the above, but in this section

#### Citation

Bai, Z. D.; Silverstein, Jack W. Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices. Ann. Probab. 27 (1999), no. 3, 1536--1555. doi:10.1214/aop/1022677458. https://projecteuclid.org/euclid.aop/1022677458

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