  We investigate the probability density of rescaled sums of iterates of
deterministic dynamical systems, a problem relevant for many complex physical
systems consisting of dependent random variables. A Central Limit Theorem (CLT)
is only valid if the dynamical system under consideration is sufficiently
mixing. For the fully developed logistic map and a cubic map we analytically
calculate the leading-order corrections to the CLT if only a finite number of
iterates is added and rescaled, and find excellent agreement with numerical
experiments. At the critical point of period doubling accumulation, a CLT is
not valid anymore due to strong temporal correlations between the iterates.
Nevertheless, we provide numerical evidence that in this case the probability
density converges to a $q$-Gaussian, thus leading to a power-law generalization
of the CLT. The above behavior is universal and independent of the order of the
maximum of the map considered, i.e. relevant for large classes of critical
dynamical systems.
