Limits of Extreme Eigenvalues

Abstract

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.