Higher Dimensions

Part of the Lecture Notes in Statistics book series (LNS, volume 71)


In this chapter the results of Chap. 3 are extended to physical systems with many degrees of freedom. Given a random vector X=(X1,..., Xn), the behavior of (iX)(mod 1) = ((tX1) (mod 1),..., (tXn)(mod 1)) is studied in detail in Sect.4.1. A necessary and sufficient condition for weak-star convergence of (tX)(mod 1) to a distribution uniform on [0, l]n, Un, as t tends to infinity, is established in Theorem 4.2 (Borel, Hopf, Kallenberg). The random vector (tX)(mod 1) converges to Un in the variation distance if and only if X has a density (Theorem 5.3).


Random Vector Variation Distance Joint Density Initial Displacement Billiard Ball 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  1. 1.Department of EconomicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Departamento de Ingeniería IndustrialUniversidad de ChileSantiagoChile

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