Abstract: 
Let $\Gamma$ be a finite additive group. An $m$composition over $\Gamma$ is an $m$tuple $(g_1,g_2,\ldots,g_m)$ over $\Gamma$.
It is called an $m$composition of $g$ if $\sum_{j=1}^m g_j = g$. A composition $(g_j)$ over $S$ is called {\em locally restricted} if there is a positive integer $\sigma$
such that any $\sigma$ consecutive parts of $(g_j)$ satisfy certain conditions. Locally restricted compositions over $\Gamma$ are associated with walks in a de Bruijin graph.
Under certain aperiodic conditions, we will show that the asymptotic number of $m$compositions of $\gamma$ is independent of $\gamma$.
We also show that the distribution of the number of occurrences of a set of subwords in such $m$compositions
is asymptotically normal with mean and variance proportional to $m$. The proofs use the transfer matrix, Kronecker product of matrices, and PerronFrobenius theorem.

