  A sharp version of the Central Limit Theorem for linear combinations of
iterates of an inner function is proved. The authors previously showed this
result assuming a suboptimal condition on the coefficients of the linear
combination. Here we explain a variation of the original argument which leads
to the sharp result. We also review the steps of the proof as well as the main
technical tool, which is Aleksandrov Desintegration Theorem for
Aleksandrov-Clark measures.
