THE INVERSION FORMULA OF THE STIELTJES TRANSFORM OF SPECTRAL DISTRIBUTION

  • Received : 2009.06.24
  • Accepted : 2009.08.20
  • Published : 2009.09.30

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 $B_{n}\;=\;\frac{1}{n}Y_{m}^{T}T_{m}Y_{m}$ where $Ym\;=\;[Y_{ij}]_{m{\times}n}$ is with independent, identically distributed entries and $T_m$ is an $m{\times}m$ symmetric nonnegative definite random matrix independent of the $Y_{ij}{^{\prime}}s$. In the present paper, using the inversion formula of the Stieltjes transform, we will find the density function of the limiting distribution of $B_n$ away from zero.

Keywords

References

  1. V. A. Marcenko and L. A. Pastur, Distribution of eignvalues for some sets of random matrices, Mathematics of the USSR-Sbornik 1 (1967), no. 4, 457-483. https://doi.org/10.1070/SM1967v001n04ABEH001994
  2. J. W. Silverstein and S. I. Choi, Analysis of limiting spectral distribution function of large dimensional random matrices, Journal of Multivariate Analysis l54 (1995), no. 2, 295-309. https://doi.org/10.1006/jmva.1995.1058
  3. 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.
  4. Y. Q. Yin, Limiting spectral distribution for a class of random matrices, Journal of Multivariate Analysis 20 (1986), no. 1, 50-68 https://doi.org/10.1016/0047-259X(86)90019-9