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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).
KeywordsRandom Vector Variation Distance Joint Density Initial Displacement Billiard Ball
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