i have a question from a mid session exam. I was marked incorrect and I dont understand why. Question: Given a reversible markov chain P, with measure $\pi$. Show that $P^2$ is reversible.

Since P is reversible $\pi$ is the invariant measure. Since $\pi$ is the invariant measure $\pi P = \pi$. Therefore, $\pi P^2=\pi PP =\pi P$. Which is reversible given from the question.

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    $\begingroup$ This shows that $\pi$ is invariant, not that the chain is reversible. $\endgroup$ – Xi'an Apr 9 at 9:18
  • $\begingroup$ So we cant reduce the terms in this way to show that its reversible too? $\endgroup$ – monraf Apr 9 at 9:26
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    $\begingroup$ Kolmogorov's Criterion offers one approach to a solution. $\endgroup$ – whuber Apr 9 at 13:45
  • $\begingroup$ Just apply the definition that $\pi_i p_{ij}=\pi_j p_{ji}$. $\endgroup$ – Xi'an Apr 10 at 14:28
  • $\begingroup$ So honestly the reason i didnt use that way is i panicked and forgot where i and j go, so i used what i thought may be a seperate method. I understand it's not the usual way i just dont understand why its wrong...i think you're saying im proving something about the measure rather than the transition matrix. I guess the real question is does invariance imply reversible as reversible implies invariance? $\endgroup$ – monraf Apr 10 at 21:36

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