Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

THE CONTINUOUS DENSITY FUNCTION OF THE LIMITING SPECTRAL DISTRIBUTION

  • Published : 2010.01.30
Download PDF
⟨ Previous Next ⟩

Abstract

In multivariate analysis, the inversion formula of the Stieltjes transform is used to find the density of a spectral distribution of random matrices of sample covariance type. Let $B_n\;=\;\frac{1}{N}Y_nY_n^TT_n$ where $Y_n\;=\;[Y_{ij}]_{n\;{\times}\;N}$ is with independent, identically distributed entries and $T_n$ is an $n\;{\times}\;n$ symmetric non-negative definite random matrix independent of the $Y_{ij}$'s. In the present paper, using the inversion formula of the Stieltjes transform, we will find that the limiting distribution of $B_n$ has a continuous density function away from zero.

Keywords

References

  1. D. Jonsson, Some limit theorems for the eigenvalues of a sample covariance matrix, Journal of Multivariate Analysis 12 (March 1982), 1-38. https://doi.org/10.1016/0047-259X(82)90080-X
  2. V. A. Marcenko and L. A. Pastur, Distribution of eignvalues for some sets of random matrices, Mathematics of the USSR-Sbornik 1, no. 4 (1967), 457-483. https://doi.org/10.1070/SM1967v001n04ABEH001994
  3. J. W. Silverstein, The limiting eigenvalue distribution of a multivariate F matrix, SIAM Journal on Applied Mathematics 16, no. 3 (May 1985), 641-646. https://doi.org/10.1137/0516047
  4. J. W. Silverstein and S. I. Choi, Analysis of limiting spectral distribution function of large dimensional random matrices, Journal of Multivariate Analysis 54, no. 2 (August 1995), 295-309. https://doi.org/10.1006/jmva.1995.1058
  5. J. W. Silverstein and P. L. Combettes, Spectral theory of large dimensional random matrices applied to signal detection, Tech. rep , Dept. Mathematics, North Carolina State Univ., Raleigh, NC (1990).
  6. J. W. Silverstein and P. L. Combettes, Signal detection via spectral theory of large dimensional random matrices, IEEE Trans. Signal Processing 40, no. 3 (August 1992), 2100-2105.
  7. K. W. Wachter, The strong limits of random matrix spectra for sample matrices of independent elements, Annals of Probability 6, no. 1 (April 1985), 1-18.
  8. Y. Q. Yin, Limiting spectral distribution for a class of random matrices, Journal of Multivariate Analysis 20, no. 1 (October 1986), 50-68 https://doi.org/10.1016/0047-259X(86)90019-9