Abstract
Let ℰ be an Euclidean space; Y n , Z, U random vectors in ℰ; h n , g n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that gn \(g_n Y_n \xrightarrow{D}Z\) and \(h_n Y_n \xrightarrow{D}U\) where Z is nonsingular. The behaviour of γ n = h n g −1 n as n→∞ is discussed first. The results are used then to prove that if\(h_n Y_{[nt]} \xrightarrow{D}Z_t\) ∃ℰfor all t∃(0, ∞), where h n ∃þ and Z 1 is nonsingular and nonsymmetric with respect to þ then \(\gamma _n (t) = h_n h_{_{[nt]} }^{ - 1} \to \gamma (t)\) ∃ H, \(Z_t \mathop = \limits^D \gamma (t)Z_1\) for all t∃(0,∞) and γ is a continuous homomorphism of the multiplicative group of (0, ∞) into þ. The explicit forms of the possible γ are shown.
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Weissman, I. On convergence of types and processes in Euclidean space. Z. Wahrscheinlichkeitstheorie verw Gebiete 37, 35–41 (1976). https://doi.org/10.1007/BF00536296
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DOI: https://doi.org/10.1007/BF00536296