Some asymptotic properties of the spectrum of the Jacobi ensemble

Abstract

For the random eigenvalues with density corresponding to the Jacobi ensemble $$c\cdot \prod _{i<j}|{\lambda }_i-{\lambda }_j{|}^{\beta }\prod _{i=1}^n(2-{\lambda }_i{)}^a(2+{\lambda }_i{)}^b{I}_{(-2,2)}({\lambda }_i)$$ $(a,b>-1,\beta >0)$ 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.

20 pages, 2 figures