Part-products of random integer compositions

A composition of n is a sequence of positive integers, called parts, that sum to n. Given a set S of positive integers, we consider compositions chosen randomly from a uniform distribution on the set of all compositions of n with parts in S. Three progressively more difficult choices of S are considered: unrestricted compositions, where S = Z₊; 1-free compositions, where S = Z₊\{1}; and S-restricted compositions, where S is an arbitrary cofinite subset of Z₊. For each choice of S, we regard the product of the parts as a random variable. We begin by deriving formulas for the moments of both the part-product and its logarithm and then proceed to the more challenging problem of proving that the part-product is asymptotically lognormal. In the case of unrestricted compositions, the calculations are relatively easy to complete using classical methods. However, those methods break down for the remaining two choices of S. We therefore introduce and formalize two new techniques for studying random compositions, the "embedding" technique and the "blocking" technique, which lead to proofs of the asymptotic lognormality of the product of parts for 1-free and S-restricted compositions respectively.