  We establish necessary and sufficient conditions for convergence (in the
sense of finite dimensional distributions) of multiplicative measures on the
set of partitions. We show that this convergence is equivalent to asymptotic
independence of finite sizes of components. The multiplicative measures depict
component spectra of random structures, the equilibrium of classic models of
statistical mechanics and stochastic processes of coagulation-fragmentation. We
then apply Schur's tauberian lemma and some results from additive number theory
and enumerative combinatorics, in order to verify the conditions derived in
important special cases. Our results demostrate that the common belief that
interacting groups in mean field models become independent as the number of
particles goes to infinity, is not true in general.
