Journal of Mathematical Sciences, Vol. 218, No. 6, Month, 2016 LIMITING THEOREMS ON SOME COMPLETE GROUPS T. Bokelavadze and G. Lominashvili UDC 512.54 Abstract. The notion of a W -exponential Hall group is introduced and the limiting theorems on w-complete groups are proved. On the real straight line, in R k , and in a Banach space, the normed sums 1 n x 1 + 1 n x 2 + ··· + 1 n x n and 1 √ n y 1 + 1 √ n y 2 + ··· + 1 √ n y n , y ν = x ν − Ex ν have especially attractive limiting properties. On a locally compact group, there exists no operation that corresponds to multiplication by a scalar c · x in a linear vector space. This leads to a difficulty which will be briefly discussed here. P. Hall introduced so-called W -power groups or simply W -groups. These groups generalize unions of W -modules for the case of arbitrary nilpotent groups. The meaning of a W -group in the general abstract theory is explained by the fact that any finitely generated, nilpotent, torsion-free group is embedded in some W -group (see [5]). Recall some definitions. Let W be an integral domain of characteristic zero such that, if λ ∈ W , then  λ n  = λ(λ − 1)(λ − 2) ··· (λ − n + 1) n! ∈ W for all n =1, 2, .... Such rings are called binomial rings. Definition 1. Let G be an arbitrary locally nilpotent group and W be a binomial ring. The group G is called a W -power group (or simply W -group) if on the product G × W there is a g-valued mapping (x, λ) → x λ , x ∈ G, λ ∈ W , such that the following axioms are fulfilled: (1) x ′ = x · x λ+μ = x λ x μ · x λμ =(x λ ) μ ; (2) y −2 x λ y =(y −1 xy)x λ ; (3) x λ 1 x λ 2 ··· x λ n = t λ 1 (x 1 ,...,x n )t ( λ 2 ) 2 (x 1 ,...,x n ) ··· t ( λ c ) n (x 1 ,...,x n ), where c is a class of nilpotency of the group {x 1 ,x 2 ,...,x n } and t i (x 1 ,x 2 ,...,x n ) is the product of commutator weights, not less than i (i =1, 2,...,n), and t 1 (x 1 ,x 2 ,...,x n )= x 1 x 2 ··· x n . The fact that axioms (1) and (2) are natural, is obvious. Axiom (3) is the transfer of a similar property for integral λ to locally nilpotent groups. These axioms imply that 1 λ = 1, x 0 = 1, and x −λ =(x λ ) −1 . We can easily see that if G is abelian, then these axioms turn into an ordinary definition of a W -module in which the usual additive notions are replaced by multiplicative ones. It is obvious that the W -group is a multiplicative group. The notions of W -subgroups, W -factor- groups, and W -homomorphisms are introduced customarily. We shall say that a subgroup H is W -admissible in G if it is a W -subgroup. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applica- tions), Vol. 97, Proceedings of the International Conference “Lie Groups, Differential Equations, and Geometry,” June 10–22, 2013, Batumi, Georgia, Part 2, 2015. 1072–3374/16/2186–0719 c  2016 Springer Science+Business Media New York 719 DOI 10.1007/s10958-016-3055-z