Advertisement

Functions of a Markov Process that are Markovian

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

Summary

In this paper we are primarily concerned with discrete time parameter Markov processes {X(n)}, n = 0, 1, 2, …, with stationary transition mechanism. The processes {Y(n)} = {f(X(n))} generated by a given many-one function f and the processes {X(n)} with a fixed stationary transition mechanism are constructed. The processes {Y(n)} are in a one-to-one correspondence with the possible initial distributions of {X(n)}. The object of the paper is to determine conditions under which {Y(n)} is Markovian, whatever the initial distribution of {X(n)}. Necessary and sufficient conditions for the new processes {Y(n)} to be Markovian are obtained under the assumption that the family of measures corresponding to the fixed transition mechanism (of {X(n)}) is dominated [4]. The conditions are expressed, of course, in terms of the function f(•) and the transition mechanism. Generally the processes {Y(n)} do not have stationary transition mechanism. The conditions simplify in the case of a continuous time parameter Markov chain. Some of the discussions may at times have the flavor of those used in considering the concept of sufficiency [4]. Some aspects of the problem discussed in the paper are touched on in [2].

Keywords

Probability Measure Markov Process Transition Function Initial Distribution Transition Mechanism 
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.

References

  1. 1.
    R. R. Bahadur, Sufficiency and statistical decision functions, Ann. Math. Stat, 25 (1954), pp. 423–462.MathSciNetCrossRefGoogle Scholar
  2. 2.
    C. J. Burke & M. Rosenblatt, A Markovian function of a Markov chain, Ann. Math. Stat., 29 (1958), pp. 1112–1122MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    J. L. Doob, Stochastic Processes, New York (1952).Google Scholar
  4. 4.
    P. R. Halmos & L. J. Savage, Application of the Radon-Nikodym theorem to the theory of sufficient statistics, Ann. Math. Stat, 20 (1949), pp. 225–241.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    M. Lqeve, Probability Theory, New York (1955).Google Scholar
  6. 6.
    B. Rankin, The concept of enchainment—a relation between stochastic processes, unpublished (1955).Google Scholar
  7. 7.
    D. Rosenblatt, On aggregation and consolidation in linear systems, to be published in the Naval Research Logistics Quarterly. Google Scholar
  8. 8.
    D. Rosenblatt, On aggregation and consolidation in linear systems, Technical Report C, prepared under Contract Nonr-1180 (00) (NR-047–012) 1956.Google Scholar
  9. 9.
    D. Rosenblatt, Abstracts of “On aggregation and consolidation in finite substochastic systems I, II, III, IV,” Ann. Math. Stat, 28 (1957), pp. 1060–1061.Google Scholar
  10. 10.
    M. Rosenblatt, Abstract of “Functions of Markov Chains,” Ann. Math. Stat., 29 (1958), p. 1291.MathSciNetGoogle 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

Personalised recommendations