  We look at the limit distributions of sums of deterministic chaotic variables
in unimodal maps and find a remarkable renormalization group (RG) structure
associated to the operation of increment of summands and rescaling. In this
structure - where the only relevant variable is the difference in control
parameter from its value at the transition to chaos - the trivial fixed point
is the Gaussian distribution and a novel nontrivial fixed point is a
multifractal distribution that emulates the Feigenbaum attractor, and is
universal in the sense of the latter. The crossover between the two fixed
points is explained and the flow toward the trivial fixed point is seen to be
comparable to the chaotic band merging sequence. We discuss the nature of the
Central Limit Theorem for deterministic variables.
