Approximation of Beta-Jacobi Ensembles by Beta-Laguerre Ensembles

Consider beta-Laguerre ensembles mu with parameters m, a(1) and beta-Jacobi ensembles lambda with parameters m, a(1), a(2). With the help of tridiagonal models of beta ensembles, we are able to prove that lim(a2)->infinity d( L (2a lambda), L (mu)) = 0 if a(1)m = o(a(2)) and lim(a2)->infinity d( L (2a lambda), L (mu)) > 0 if lim(a2)->infinity (a1m)/(a2) = sigma> 0, by contrast, where a := a(1) + a(2) and d is total variation distance or Kullback{Leibler divergence. This result improves the approximation in [9].