The paper studies the limiting behaviour of spectral measures of random Jacobi matrices of Gaussian, Wishart and MANOVA beta ensembles. We show that the spectral measures converge weakly to a limit distribution which is the semicircle distribution, Marchenko-Pastur distributions or Kesten-McKay distributions, respectively. The Gaussian fluctuation around the limit is then investigated.
"On spectral measures of random Jacobi matrices." Osaka J. Math. 55 (4) 595 - 617, October 2018.