  We derive some Quantum Central Limit Theorems for expectation values of
macroscopically coarse-grained observables, which are functions of
coarse-grained hermitean operators. Thanks to the hermicity constraints, we
obtain positive-definite distribution for the expectation values of
observables. These probability distributions open some pathway for an emergence
of classical behaviours in the limit of infinitely large number of identical
and non-interacting quantum constituents. This is in contradistinction to other
mechanisms of classicality emergence due to environmental decoherence and
consistent histories. The probability distributions so derived also enable us
to evaluate the nontrivial time-dependence of certain differential entropies.
