Convergence of the empirical spectral distribution function of Beta matrices

Let $\mathbf{B}_n=\mathbf {S}_n(\mathbf {S}_n+\alpha_n\mathbf {T}_N)^{-1}$, where $\mathbf {S}_n$ and $\mathbf {T}_N$ are two independent sample covariance matrices with dimension $p$ and sample sizes $n$ and $N$, respectively. This is the so-called Beta matrix. In this paper, we focus on the limiting spectral distribution function and the central limit theorem of linear spectral statistics of $\mathbf {B}_n$. Especially, we do not require $\mathbf {S}_n$ or $\mathbf {T}_N$ to be invertible. Namely, we can deal with the case where $p>\max\{n,N\}$ and $p<n+N$. Therefore, our results cover many important applications which cannot be simply deduced from the corresponding results for multivariate $F$ matrices.

[1]  J. W. Silverstein,et al.  Spectral Analysis of Large Dimensional Random Matrices , 2009 .

[2]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[3]  Shurong Zheng,et al.  Central limit theorems for linear spectral statistics of large dimensional F-matrices , 2012 .

[4]  Z. D. Bai,et al.  On the Limiting Empirical Distribution Function of the Eigenvalues of a Multivariate F Matrix , 1988 .

[5]  P. Rousseeuw,et al.  Wiley Series in Probability and Mathematical Statistics , 2005 .

[6]  J. W. Silverstein,et al.  No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices , 1998 .

[7]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[8]  Z. Bai,et al.  CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data , 2017, Statistical Papers.

[9]  Y. Yin.  Limiting spectral distribution for a class of random matrices , 1986 .

[10]  Cedric E. Ginestet.  Spectral Analysis of Large Dimensional Random Matrices, 2nd edn , 2012 .

[11]  J. W. Silverstein,et al.  Analysis of the limiting spectral distribution of large dimensional random matrices , 1995 .

[12]  Z. D. Bai,et al.  The limiting spectral distribution of the product of the Wigner matrix and a nonnegative definite matrix , 2010, J. Multivar. Anal..

[13]  Jean-Paul Chilès,et al.  Wiley Series in Probability and Statistics , 2012 .

[14]  C. Parvin.  An Introduction to Multivariate Statistical Analysis, 3rd ed. T.W. Anderson. Hoboken, NJ: John Wiley & Sons, 2003, 742 pp., $99.95, hardcover. ISBN 0-471-36091-0. , 2004 .

[15]  J. Norris.  Appendix: probability and measure , 1997 .

[16]  E. Wigner.  Characteristic Vectors of Bordered Matrices with Infinite Dimensions I , 1955 .

[17]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[18]  Global Fluctuations for Linear Statistics of \beta-Jacobi Ensembles , 2012, 1203.6103.

[19]  Z. Bai,et al.  On Limiting Empirical Distribution Function of the Eigenvalues of a Multivariate F Matrix. Revised. , 1984 .

[20]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of a class of large dimensional random matrices , 1995 .

[21]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[22]  J. W. Silverstein.  Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices , 1995 .