Approximation of Beta-Jacobi ensembles by Beta-Laguerre ensembles

Let $\lambda$ and $\mu$ be beta-Jacobi and beta-Laguerre ensembles with joint density function $f_{\beta ,m,a_1,a_2}$ and $f_{\beta ,m,a_1}$, respectively. Here $\beta >0$ and $a_1,a_2$ and $m$ satisfying . $a_1, a_2>\frac{\beta}{2}(m-1).$Inthispaper,weconsiderthedistancebetween$2(a_1+a_2)\lambda$and$\mu$ in terms of total variation distance and Kullback-Leibler distance. Following the idea in \cite{JM2017}, we are able to prove that both the two distances go to zero once $a_{1}m=o(a_2)$ and not so if ${\lim }_{a_2\to \mathrm{\infty }}{a}_{1}m\mathrm{/}{a}_2=\sigma >0.$