  Two different aspects of parabolic iteration in the complex upper half-plane
are considered here. First, from a noncommutative probability perspective, a
Berry-Esseen type estimate for the convergence speed of the monotone central
limit theorem is proved. Secondly, if the underlying measure in this central
limit process is singular to the Lebesgue measure on the real line, then the
iteration is shown to be an infinite-measure preserving dynamical system that
has a regularly varying return sequence of index 1/2.
