  We show that for many families of OPUC, one has $||\varphi'_n||_2/n -> 1$, a
condition we call normal behavior. We prove that this implies $|\alpha_n| -> 0$
and that it holds if the sequence $\alpha_n$ is in $\ell^1$. We also prove it
is true for many sparse sequences. On the other hand, it is often destroyed by
the insertion of a mass point.
