Advertisement

Remarks on ergodicity of stationary irreducible transient Markov chains

  • M. Rosenblatt
Article
  • 71 Downloads

Summary

Consider a stationary process {Xn(Ω), − ∞ < n < ∞. If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {Xn(Ω), -∞ < n < ∞ and of each of the subprocesses {Xn(Ω), 0 ≦n < ∞, {Xn(Ω), −∞ < n ≦ 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {Xn(Ω), - ∞ < n <∞} with {itXn(Ω), 0 ≦n < <∞ ergodic but {Xn(Ω), ∞ < n ≦0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {Xn(Ω), 0 ≦n < ∞ and {itX n(Ω),-<∞ < n ≦ 0} ergodic but {Xn(Ω), -∞ < n < ∞ nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {Xn(Ω), -∞ < n < ∞< are nonergodic.

Keywords

Markov Chain Stochastic Process Stationary Process Probability Theory Mathematical Biology 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Blackwell, D.: On transient Markov processes with a countable number of states and stationary transition probabilities. Ann. math. Statistics 26, 654–658 (1955).Google Scholar
  2. [2]
    Chung, K. L.: Markov chains with stationary transition probabilities. Berlin-Göttingen-Heidelberg: Springer 1960.Google Scholar
  3. [3]
    Doob, J. L.: Stochastic processes. New York: Wiley 1953.Google Scholar
  4. [4]
    Hunt, G. A.: Markoff chains and Martin boundaries. Illinois J. Math. 4, 313–340 (1960).Google Scholar
  5. [5]
    Isaacs, R.: A uniqueness theorem for stationary measures of ergodic Markov processes. Ann. math. Statistics 34, 1781–1785 (1964).Google Scholar
  6. [6]
    Kakutani, S., and W. Parry: Infinite measure preserving transformations with “mixing”. Bull. Amer. math. Soc. 69, 752–756 (1963).Google Scholar
  7. [7]
    Veech, W.: The necessity of Harris' condition for existence of a stationary measure. Proc. Amer. math. Soc. 14, 856–860 (1963).Google Scholar

Copyright information

© Springer-Verlag 1966

Authors and Affiliations

  • M. Rosenblatt
    • 1
    • 2
  1. 1.University of CaliforniaSan Diego
  2. 2.University College LondonUSA

Personalised recommendations