We establish necessary and sufficient conditions for the convergence (in the sense of
finite dimensional distributions) of multiplicative measures on the set of partitions. The
multiplicative measures depict distributions of component spectra of random structures and
also the equilibria of classic models of statistical mechanics and stochastic processes of
coagulation-fragmentation. We show that the convergence of multiplicative measures is
equivalent to the asymptotic independence of counts of components of fixed sizes in random
structures. We then apply Schur’s tauberian lemma and some results from additive number
theory and enumerative combinatorics in order to derive plausible sufficient conditions of
convergence. Our results demonstrate that the common belief, that counts of components of
fixed sizes in random structures become independent as the number of particles goes to
infinity, is not true in general.