Consider a finite Markov chain with exactly $m$ states. Is it possible for any state $i$ to have a finite nonzero period $n > m$? Any help would be greatly appreciated.
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No, suppose we have the shortest path from $i$ back to $i$ with $n$ edges. If $n > m$ then by the pigeonhole principle we must visit some state twice on this path. However since Markov chains are memoryless these two visits are not distinguishable and thus the second visit is superfluous, and one visit could've sufficed. This breaks our assumption that we had the shortest path and $n > m$, and thus the shortest path (if it exists) must have $n \leq m$ edges.
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I got it, thank you! – Ageron Jun 05 '21 at 15:57