  Pickrell has fully characterized the unitarily invariant probability measures
on infinite Hermitian matrices, and an alternative proof of this classification
has been found by Olshanski and Vershik. Borodin and Olshanski deduced from
this proof that under any of these invariant measures, the extreme eigenvalues
of the minors, divided by the dimension, converge almost surely. In this paper,
we prove that one also has a weak convergence for the eigenvectors, in a sense
which is made precise. After mapping Hermitian to unitary matrices via the
Cayley transform, our result extends a convergence proven in our paper with
Maples and Nikeghbali, for which a coupling of the Circular Unitary Ensemble of
all dimensions is considered.
