  We study of the effect of weak noise on critical one dimensional maps; that
is, maps with a renormalization theory.
  We establish a one dimensional central limit theorem for weak noises and
obtain Berry--Esseen estimates for the rate of this convergence.
  We analyze in detail maps at the accumulation of period doubling and critical
circle maps with golden mean rotation number. Using renormalization group
methods, we derive scaling relations for several features of the effective
noise after long times. We use these scaling relations to show that the central
limit theorem for weak noise holds in both examples.
  We note that, for the results presented here, it is essential that the maps
have parabolic behavior. They are false for hyperbolic orbits.
