Limits of Extreme Eigenvalues

  • Zhidong Bai
  • Jack W. Silverstein
Part of the Springer Series in Statistics book series (SSS)


In multivariate analysis, many statistics involved with a random matrix can be written as functions of integrals with respect to the ESD of the random matrix. When the LSD is known, one may want to apply the Helly-Bray theorem to find approximate values of the statistics. However, the integrands are usually unbounded. For instance, the integrand in Example 1.2 is log x, which is unbounded both from below and above. Thus, one cannot use the LSD and Helly-Bray theorem to find approximate values of the statistics. This would render the LSD useless. Fortunately, in most cases, the supports of the LSDs are compact intervals. Still, this does not mean that the Helly-Bray theorem is applicable unless one can prove that the extreme eigenvalues of the random matrix remain in certain bounded intervals.


Random Matrix Large Eigenvalue Sample Covariance Matrix Fourth Moment Quaternion Matrix 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Mathematics and Statistics KLAS MOE Northeast Normal UniversityChangchunChina
  2. 2.Department of Statistics and Applied ProbabilityNational University of SingaporeSingaporeSingapore
  3. 3.Department of MathematicsNorth Carolina State UniversityRaleighUS

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