  In this paper, we study the extremal process of the maxima of all the largest
eigenvalues of principal minors of the classical Gaussian orthogonal ensemble
(GOE). We prove that the fluctuation of the maxima is given by the Gumbel
distribution in the limit. We also derive the limiting joint distribution of
the maxima and the corresponding eigenvector, which implies that these two
random variables are asymptotically independent.
