Nonstandard limit theorems and large deviation for beta -Jacobi ensembles with a different scaling

We consider $\beta$-Jacobi ensembles with parameters $p_1,p_2\ge n\mathrm{.}$ We prove that the empirical measure of the rescaled Jacobi ensembles converges weakly to a modified Watcher law via the spectral measure method, which revisits the weak limits obtained in \cite{MaLDPJ} while replacing the condition $\beta n\text{\hspace{0.17em}⁣}>\text{\hspace{0.17em}⁣}>\log n$ by $\beta n\text{\hspace{0.17em}⁣}>\text{\hspace{0.17em}⁣}>1.$ We also provide the central limit theorem and the large deviation for the corresponding rescaled spectral measure.