Some asymptotic properties of the spectrum of the Jacobi ensemble

TL;DR

This paper studies the asymptotic behavior of the spectrum of the Jacobi ensemble, providing uniform approximations of eigenvalues by roots of Jacobi polynomials and analyzing spectral distribution as parameters grow with sample size.

Contribution

It introduces a strong uniform approximation of eigenvalues by Jacobi polynomial roots with error bounds, extending understanding of spectral properties in large-dimensional settings.

Findings

Eigenvalues can be approximated by roots of Jacobi polynomials with explicit error bounds.

Asymptotic spectral distribution analyzed as parameters grow with sample size.

Applications discussed in multivariate random F-matrix context.

Abstract

For the random eigenvalues with density corresponding to the Jacobi ensemble $$c \cdot \prod_{i < j} | \lambda_i - \lambda_j |^\beta \prod^n_{i=1} (2 - \lambda_i)^a (2 + \lambda_i)^b I_{(-2,2)} (\lambda_i) $$ $(a, b > -1, \beta > 0) $ a strong uniform approximation by the roots of the Jacobi polynomials is derived if the parameters $a, b,$ $\beta$ depend on $n$ and $n \to \infty$. Roughly speaking, the eigenvalues can be uniformly approximated by roots of Jacobi polynomials with parameters $((2a+2)/\beta -1, (2b+2)/\beta-1)$, where the error is of order $\{\log n/(a+b) \}^{1/4}$. These results are used to investigate the asymptotic properties of the corresponding spectral distribution if $n \to \infty$ and the parameters $a, b$ and $\beta$ vary with $n$. We also discuss further applications in the context of multivariate random $F$-matrices.