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Recently I'm reading Markov chains and Stochastics Stability(sencond edition 2009) written by Meyn and Tweedie. And in the proof of Theorem 10.4.5 on page 243, it says "if $\pi_m$ is invariant for the $m$-skeleton then by aperiodicity the measure is the unique invariant measure(up to multiples) for $\Phi^m$." is that right? If not, is there any counter-example?

Here is the Theorem 10.4.5:

Suppose that Markov Chain $\Phi$ is $\psi$-irreducible and aperiodic. Then, for each $m$, a measure $\pi$ is invariant for the $m$-skeleton if and only if it is invariant for $\Phi$.

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    $\begingroup$ What's $m$-skeleton? $\endgroup$ – R W Dec 15 '13 at 5:16
  • $\begingroup$ $\Phi_0, \Phi_m, \Phi_{2m}, \Phi_{3m}, ...$ picked out from the original chain $\endgroup$ – Wieshawn Dec 15 '13 at 6:08

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