We derive some Quantum Central Limit Theorems for expectation values of macroscopically
coarse-grained observables, which are functions of coarse-grained hermitean operators consisting
of non-commuting variables. Thanks to the hermicity constraints, we obtain positive-definite dis-
tribution for the expectation values of observables. These probability distributions open some
pathway for an emergence of classical behaviours in the limit of innitely 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 probabil-
ity distributions so derived also enable us to evaluate the nontrivial time-dependence of certain
differential entropies.