Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices

نویسنده

  • Jack W. Silverstein
چکیده

Let B n = (1/N)T 1/2 n is a Hermitian square root of the nonnegative definite Hermitian matrix T n. It is shown in Bai and Silverstein (1998) that, under certain conditions on the eigenvalues of T n , with probability one 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 one the number of eigenvalues of B n and T n lying on one side of their respective intervals are identical for all large n.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Spectrum estimation for large dimensional covariance matrices using random matrix theory

Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics; the eigenvalues of covariance matrices play a key role in many widely techniques, in particular in Principal Component Analysis (PCA). In many modern data analysis problems, statisticians are faced with large datasets where the sample si...

متن کامل

Estimating principal components of large covariance matrices using the Nyström method

Covariance matrix estimates are an essential part of many signal processing algorithms, and are often used to determine a lowdimensional principal subspace via their spectral decomposition. However, for sufficiently high-dimensional matrices exact eigenanalysis is computationally intractable, and in the case of limited data, sample eigenvalues and eigenvectors are known to be poor estimators of...

متن کامل

Large Deviations for Eigenvalues of Sample Covariance Matrices, with Applications to Mobile Communication Systems

We study sample covariance matrices of the form W = (1/n)CC , where C is a k × n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-calledWishart matrices, where the entries ofC are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, ...

متن کامل

Operator norm consistent estimation of large dimensional sparse covariance matrices

Estimating covariance matrices is a problem of fundamental importance in multivariate statistics. In practice it is increasingly frequent to work with data matrices X of dimension n × p, where p and n are both large. Results from random matrix theory show very clearly that in this setting, standard estimators like the sample covariance matrix perform in general very poorly. In this “large n, la...

متن کامل

Consistent Estimation of Large - Dimensional Sparse Covariance Matrices

Estimating covariance matrices is a problem of fundamental importance in multivariate statistics. In practice it is increasingly frequent to work with data matrices X of dimension n×p, where p and n are both large. Results from random matrix theory show very clearly that in this setting, standard estimators like the sample covariance matrix perform in general very poorly. In this “large n, larg...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 1999