Part-products of S-restricted integer compositions

Abstract

If S is a cofinite set of positive integers, an "S-restricted composition of n" is a sequence of elements of S, denoted ?? = (?1, ?2, ... ), whose sum is n. For uniform random S-restricted compositions, the random variable B(??) = ?i ?i is asymptotically lognormal. (A precise statement of the theorem includes an error term to bound the rate of convergence.) The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.