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On convergence of types and processes in Euclidean space

  • Ishay Weissman
Article

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 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.

Keywords

Stochastic Process Probability Theory Euclidean Space Explicit Form Random Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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