# Reversible markov chain

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.

• This shows that $\pi$ is invariant, not that the chain is reversible. – Xi'an Apr 9 at 9:18
• So we cant reduce the terms in this way to show that its reversible too? – monraf Apr 9 at 9:26
• Kolmogorov's Criterion offers one approach to a solution. – whuber Apr 9 at 13:45
• Just apply the definition that $\pi_i p_{ij}=\pi_j p_{ji}$. – Xi'an Apr 10 at 14:28
• 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? – monraf Apr 10 at 21:36