I'm trying to prove that if
$X_n$ iid normal
$S_n = \sum_1^n X_i$
then $U_n$ is asymptotically uniform in distribution. I've got no idea how to approach this, and it's a past exam question, so I should be able to do it inside 30 mins. My only ideas for how to approach it were
(1) possibly use an ergodic theorem (since the floor function looked inviting).
(2) work with characteristic functions and use Levy's Continuity Theorem (but I can't see how to work out the characteristic functions of the $U_n$).
Has anyone got an idea how to proceed? A hint rather than a full solution would be ideal, since I'd quite like to have something to work out myself, once I've got an idea how to start!