## The Annals of Probability

### No eigenvalues outside the support of the limiting spectral distribution 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 is known that, as $n \to \infty$, if $n/N$ converges to a positive number and the empirical distribution of the eigenvalues of $T_n$ converges to a proper probability distribution, then the empirical distribution of the eigenvalues of $B_n$ converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of $T_n$, for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all $n$ sufficiently large.

#### Article information

Source
Ann. Probab., Volume 26, Number 1 (1998), 316-345.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855421

Digital Object Identifier
doi:10.1214/aop/1022855421

Mathematical Reviews number (MathSciNet)
MR1617051

Zentralblatt MATH identifier
0937.60017

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

#### Citation

Bai, Z. D.; Silverstein, Jack W. No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 (1998), no. 1, 316--345. doi:10.1214/aop/1022855421. https://projecteuclid.org/euclid.aop/1022855421

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