On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y_n, Z, U random vectors in ; h_n, g_naffine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h_ng_n^–1as n is discussed first. The results are used then to prove that if for all t(0, ), where h_nþ and Z₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.