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Stationary Processes as Shifts of Functions of Independent Random Variables

  • Richard A. Davis
  • Keh-Shin Lii
  • Dimitris N. Politis
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Summary

Let xn, n = 0, ±1, ±2, …, be a strictly stationary process. Two closely related problems are posed with respect to the structure of strictly stationary processes. In the first problem we ask whether one can construct a random variable ξn = g(xn, xn-1, …), a function of xn, xn-1, …, that is independent of the past, that is, independent of xn-1, xn-2, …. Such a sequence of random variables {ξn} is a sequence of independent and identically distributed random variables. Further, given such a construction, is xn a function of ξn, ξn-1, …. Necessary and sufficient conditions for such a representation are obtained in the case where xn is a finite state Markov chain with the positive transition probabilities in any row of the transition probability matrix P = (pii) of xn distinct (Section 3). Such a representation is comparatively rare for a finite state Markov chain. In the second problem, the assumption that the independent and identically distributed ξn’s be functions of xn, xn-1, … is removed. We ask whether for some such family {ξn} there is a process {yn}, yn = g(ξn, ξn-1,…), with the same probability structure as {xn}. This is shown to be the case for every ergodic finite state Markov chain with nonperiodic states (Section 4). Sufficient conditions for such representations in the case of a general strictly stationary process are obtained in Section 5.

Keywords

Markov Chain Transition Probability Matrix Borel Function Conditional Probability Distribution State Markov Chain 
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References

  1. 1.
    J. L. Doob, Stochastic Processes, New York (1953).Google Scholar
  2. 2.
    W. Feller, An Introduction to Probability Theory and its Applications, New York (1950).Google Scholar
  3. 3.
    G. Kallianpur & N. Weiner, Nonlinear prediction, Technical Report No. 1 (1956), Office of Naval Research, Cu-2–56-Nonr-266, (39)-CIRMIP, Project NR-047–015.Google Scholar
  4. 4.
    A. I. Khinchin, Mathematical Foundations of Information Theory, New York (1957).Google Scholar
  5. 5.
    P. Levy, Theorie de Vaddition des variables aléatoires, Paris (1937).Google Scholar
  6. 6.
    M. Loeve, Probability Theory, New York (1955).Google Scholar
  7. 7.
    M. Rosenblatt, Functions of a Markov process that are Markovian, Journal of Mathematics and Mechanics, 8 (1959), pp. 585–596.MathSciNetMATHGoogle Scholar
  8. 8.
    H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Zeit., 52 (1950), pp. 642–648.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Richard A. Davis
    • 1
  • Keh-Shin Lii
    • 2
  • Dimitris N. Politis
    • 3
  1. 1.Department of StatisticsColumbia UniversityNew YorkUSA
  2. 2.Department of StatisticsUniversity of CaliforniaRiversideUSA
  3. 3.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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