  As a counterpart to our previous study of the stationary distribution formed
by sums of positions at the Feigenbaum point via the period-doubling cascade in
the logistic map (Eur. Phys. J. B 87 32, (2014)), we determine the family of
related distributions for the accompanying cascade of chaotic band-splitting
points in the same system. By doing this we rationalize how the interplay of
regular and chaotic dynamics gives rise to either multiscale or gaussian limit
distributions. As demonstrated before (J. Stat. Mech. P01001 (2010)), sums of
trajectory positions associated with the chaotic-band attractors of the
logistic map lead only to a gaussian limit distribution, but, as we show here,
the features of the stationary multiscale distribution at the Feigenbaum point
can be observed in the distributions obtained from finite sums with
sufficiently small number of terms. The multiscale features are acquired from
the repellor preimage structure that dominates the dynamics toward the chaotic
attractors. When the number of chaotic bands increases this hierarchical
structure with multiscale and discrete scale-invariant properties develops.
Also, we suggest that the occurrence of truncated q-gaussian-shaped
distributions for specially prescribed sums are t-Student distributions
premonitory of the gaussian limit distribution.
