Total variation approximation for random assemblies and a functional limit theorem

A class of random weakly logarithmic combinatorial assemblies is explored in the paper. We extend total variation approximations for the distribution of component vector of a random structure. That leads to the probabilistic approach suitable to examine the asymptotic value distribution of additive functions defined on such assemblies with the component sizes restricted to a given set. The results generalize several investigations of random Λ-permutations and their extensions to other structures obtained mainly by the Russian mathematicians. Instead of the most popular approach based upon the Tauber type theorems, we develop a comparative asymptotical analysis of coefficients of two Taylor series. Demonstrating possible applications, we obtain necessary and sufficient conditions for the weak convergence of processes defined via partial sums of an additive function to the Brownian motion.