  We generalize Huberman-Rudnick universal scaling law for all periodic windows
of the logistic map and show the robustness of $q$-Gaussian probability
distributions in the vicinity of chaos threshold. Our scaling relation is
universal for the self-similar windows of the map which exhibit period-doubling
subharmonic bifurcations. Using this generalized scaling argument, for all
periodic windows, as chaos threshold is approached, a developing convergence to
$q$-Gaussian is numerically obtained both in the central regions and tails of
the probability distributions of sums of iterates.
