On convergence of types and processes in Euclidean space

  • Ishay Weissman


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 n −1 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 Z1 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.


Stochastic Process Probability Theory Euclidean Space Explicit Form Random Vector 
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Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Ishay Weissman
    • 1
  1. 1.Faculty of Industrial and Management EngineeringTechnionIsrael

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