The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Olshanski and Vershik. In their work, and another work by Borodin and Olshanski, the authors prove the almost sure convergence of the renormalized extreme eigenvalues of the minors when the dimension goes to infinity. In this talk, we present these results, and another convergence result on eigenvectors we prove in a work in preparation. These results apply in particular to the image of the Circular Unitary Ensemble by the Cayley transform which maps the unit circle to the real line.