Part-products of $S$-restricted integer compositions

If $S$ is a cofinite set of positive integers, an "$S$-restricted composition of $n$" is a sequence of elements of $S$, denoted $\vec{\lambda}=(\lambda_1,\lambda_2,...)$, whose sum is $n$. For uniform random $S$-restricted compositions, the random variable ${\bf B}(\vec{\lambda})=\prod_i \lambda_i$ is asymptotically lognormal. The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.