Approximation of Beta-Jacobi ensembles by Beta-Laguerre ensembles

TL;DR

This paper investigates how closely beta-Jacobi ensembles can be approximated by beta-Laguerre ensembles by analyzing their distributional distances, with results depending on the growth rates of parameters.

Contribution

It provides new theoretical bounds on the total variation and Kullback-Leibler distances between scaled beta-Jacobi and beta-Laguerre ensembles under specific parameter regimes.

Findings

Distances tend to zero if a_1m=o(a_2)

Distances do not tend to zero if a_1m/a_2 approaches a positive constant

The results extend previous work on ensemble approximation in random matrix theory.

Abstract

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 . $a_1, a_2>\frac{\beta}{2}(m-1).$ In this paper, we consider the distance between $2(a_1+a_2)\lambda$ 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 $a_1m=o(a_2)$ and not so if $\lim_{a_2\to\infty}a_1m/a_2=\sigma>0.$