Approximation of Beta-Jacobi Ensembles by Beta-Laguerre Ensembles

Consider beta-Laguerre ensembles μ with parameters m, a1 and beta-Jacobi ensembles γ with parameters m, a1, a2. With the help of tridiagonal models of beta ensembles, we are able to prove that lima2→∞d(L(2aλ),L(μ))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lim _{{a_2} \to \infty }}d\left( {{\cal L}\left( {2a{\bf{\lambda }}} \right),{\cal L}\left( {\bf{\mu }} \right)} \right) = 0$$\end{document} if a1m=o(a2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a_1}m = o\left( {{a_2}} \right)$$\end{document} and lim_a2→∞d(L(2aλ),L(μ))>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\underline {\lim } _{{a_2} \to \infty }}\,d\left( {{\cal L}\left( {2a{\bf{\lambda }}} \right),{\cal L}\left( {\bf{\mu }} \right)} \right) > 0$$\end{document} if lima2→∞a1ma2=σ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lim _{{a_2} \to \infty }}\,{{{a_1}m} \over {{a_2}}} = \sigma > 0$$\end{document}, by contrast, where a ≔ a1 + a2 and d is total variation distance or Kullback—Leibler divergence. This result improves the approximation in [9].