Weak convergence of distributions of separable statistics

This paper focuses on weak limits (as the number of summands grows) of distributions of separable statistics taking values in locally compact Hausdorff Abelian groups. We consider a general method of deriving limit theorems for separable statistics that is applicable to any schemes. The weak limits under study are expressed in terms of the limit conditional distribution of an additive form associated with the separable statistic and determined from the independent random variables.