Summary
Consider a stationary process {X n(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 {X n(Ω), -∞ < n < ∞ and of each of the subprocesses {X n(Ω), 0 ≦n < ∞, {X n(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 {X n(Ω), - ∞ < n <∞} with {itXn(Ω), 0 ≦n < <∞ ergodic but {X n(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 {X n(Ω), 0 ≦n < ∞ and {itX n(Ω),-<∞ < n ≦ 0} ergodic but {X n(Ω), -∞ < n < ∞ nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X n(Ω), -∞ < n < ∞< are nonergodic.
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This research was supported in part by the Office of Naval Research.
John Simon Guggenheim Memorial Fellow.
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Rosenblatt, M. Remarks on ergodicity of stationary irreducible transient Markov chains. Z. Wahrscheinlichkeitstheorie verw Gebiete 6, 293–301 (1966). https://doi.org/10.1007/BF00537828
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DOI: https://doi.org/10.1007/BF00537828