A Family of Two Dimensionally Determined Ergodic Processes

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• 作者：D. Hamdan (1)
  
  1. Laboratoire de Probabilit茅s et Mod猫les Al茅atoires ; CNRS-UMR 7599 ; Universit茅 Paris VI et Universit茅 Paris VII ; Tour 56 (3e 茅tage) ; Bo卯te courrier 188 ; 4 Place Jussieu ; 75752 ; Paris Cedex 05 ; France
  
• 关键词：Ergodic processes ; Group rotation ; Finite dimensional laws ; Fourier ; Stieltjes coefficients ; 42A16 ; 37A05 ; 37A50 ; 60G10
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：21
• 期：1
• 页码：137-160
• 全文大小：399 KB
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  12. Hamdan, D.: An ergodic Markov chain is not determined by any line-equation id-i-eq988"> \(p\) -marginals. Indag. Math. N. S. 13(4), 487鈥?98 (2002) get="_blank" title="It opens in new window">CrossRef
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Abstract Harmonic Analysis
  Approximations and Expansions
  Partial Differential Equations
  Applications of Mathematics
  Signal,Image and Speech Processing
  
• 出版者：Birkh盲user Boston
• ISSN：1531-5851

文摘

We prove that for an ergodic rotation \(R\) and square integrable function \(f\) on a compact abelian group, the ergodic process \(X=(f\circ R^n)_{n\in {\mathbb {Z}}}\) is uniquely determined by its two-dimensional laws if the same holds for the process \(Y=(h\circ f \circ R^n )_{n\in {\mathbb {Z}}},\) for some real bounded function \(h,\) such that all Fourier-Stieltjes coefficients of \(h\circ f\) are non null. Applied to the one or two dimensional torus, this result gives a large class of such processes, for instance any process given by non constant monotone continuous function, or having a discontinuity at an irrational point, on the unit interval, is in the corresponding class. We also prove that all Fourier coefficients of such a monotone function are non null.