  Using intermittent maps with infinite invariant measures, we investigate the
universality of time-averaged observables under aging conditions. According to
Aaronson-Darling-Kac theorem, in non-aged dynamical systems with infinite
invariant measures, the distribution of the normalized time averages of
integrable functions converge to the Mittag-Leffler distribution. This well
known theorem holds when the start of observations coincides with the start of
the dynamical processes. Introducing a concept of the aging limit where the
aging time $t_a$ and the total measurement time $t$ goes to infinity while the
aging ratio $t_a/t$ is a constant, we obtain a novel distributional limit
theorem of time-averaged observables integrable with respect to the infinite
invariant density. Applying the theorem to the Lyapunov exponent in
intermittent maps, we find that regular motions and a weakly chaotic behavior
coexist in the aging limit. This mixed type of dynamics is controlled by the
aging ratio and hence is very different from the usual scenario of regular and
chaotic motions in Hamiltonian systems. The probability of finding regular
motions in non-aged processes is zero, while in the aging regime it is finite
and it increases when system ages.
