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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 $\frak{sl}_ 2$ . The limit is taken over linearly scaled fusion powers of a fixed collection of irreducible repre sentations. This includes as special instances all Demazure modules of the affine Kac-Moody algebra associated to $\frak{ sl}_ 2$ . Along with an available complementary result on the asymptotic normality of the basic special ization 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.

Distributions Defined by $q$-Supernomials, Fusion Products, and Demazure Modules