  We prove asymptotic normality of the distributions defined by q-supernomials,
which implies asymptotic normality of the distributions given by the central
string functions and the basic specialization of fusion modules of the current
algebra of sl2. The limit is taken over linearly scaled fusion powers of a
fixed collection of irreducible representations. This includes as special
instances all Demazure modules of the affine Kac-Moody algebra associated to
sl2. Along with an available complementary result on the asymptotic normality
of the basic specialization of graded tensors of the type A standard
representation, our result is a central limit theorem for a serious class of
graded tensors. It therefore serves as an indication towards universal
behavior: The central string functions and the basic specialization of fusion
and, in particular, Demazure modules behave asymptotically normal, as the
number of fusions scale linearly in an asymptotic parameter, N say.
