Limit theorems for beta-Jacobi ensembles
نویسنده
چکیده
For a β-Jacobi ensemble determined by parameters a1, a2 and n, under the restriction that the three parameters go to infinity with n and a1 being of small orders of a2, we obtain some limit theorems about the eigenvalues. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the central limit theorems of the eigenvalues, and the limiting distributions of the empirical distributions of the eigenvalues.
منابع مشابه
Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles
We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If n is the size of the sample, r ≤ n the number of variates and Xn,r such a matrix, a generalization of the Bartlett-type theorems gives a decomposi...
متن کاملDistributions of the Extreme Eigenvaluesof Beta-Jacobi Random Matrices
We present explicit formulas for the distributions of the extreme eigenvalues of the β–Jacobi random matrix ensemble in terms of the hypergeometric function of a matrix argument. For β = 1, 2, 4, these formulas specialize to the well-known real, complex, and quaternion Jacobi ensembles, respectively.
متن کاملSome Results for the Jacobi-Dunkl Transform in the Space $L^{p}(mathbb{R},A_{alpha,beta}(x)dx)$
In this paper, using a generalized Jacobi-Dunkl translation operator, we obtain a generalization of Titchmarsh's theorem for the Dunkl transform for functions satisfying the Lipschitz Jacobi-Dunkl condition in the space Lp.
متن کاملApproximation of Haar Distributed Matrices and Limiting Distributions of Eigenvalues of Jacobi Ensembles
We develop a tool to approximate the entries of a large dimensional complex Jacobi ensemble with independent complex Gaussian random variables. Based on this and the author’s earlier work in this direction, we obtain the Tracy-Widom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko-Pastur law, the central limit theorem, and the laws of large nu...
متن کاملDouble scaling limit for matrix models with non analytic potentials
We study the double scaling limit for unitary invariant ensembles of randommatrices with non analytic potentials and find the asymptotic expansion for the entries of the corresponding Jacobi matrix. Our approach is based on the perturbation expansion for the string equations. The first order perturbation terms of the Jacobi matrix coefficients are expressed through the Hastings-McLeod solution ...
متن کامل