Remarks on ergodicity of stationary irreducible transient Markov chains

  • M. Rosenblatt
Article
  • 67 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 

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References

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Copyright information

© Springer-Verlag 1966

Authors and Affiliations

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

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