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I would like to show this assumption implies that the Markov chain is irreducible and aperiodic. I was able to show the irreducible part but I'm having trouble proving the aperiodicity. I tried to prove there must exist an odd cycle, but I couldn't seem to proceed. Any help?

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Aperiodicity is immediate from the fact if a set $A$ of integers contains two consecutive integers then its gcd is $1$. Note that $P^{r+1}$ also has all entries positive as shown below:

Suppose $M$ is a stochastic matrix . Suppose some element $\sum_k m_{ik} P^{r}_{kj}$ of the product $MP^{r}$ is $0$. Then $m_{ik}=0$ for all $k$ contradicting the fact that $\sum_k m_{ik}=1$. Apply this with $M=P$.

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