  We determine the limit distributions of sums of deterministic chaotic
variables in unimodal maps assisted by a novel renormalization group (RG)
framework associated to the operation of increment of summands and rescaling.
In this framework the difference in control parameter from its value at the
transition to chaos is the only relevant variable, the trivial fixed point is
the Gaussian distribution and a nontrivial fixed point is a multifractal
distribution with features similar to those of the Feigenbaum attractor. The
crossover between the two fixed points is discussed and the flow toward the
trivial fixed point is seen to consist of a sequence of chaotic band mergers.
