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Let $X(n)_{n \geq 0}$ be a homogeneous Markov chain with state space $S$. Define $$p_{ij}(n):= \mathbb{P}( X(n+m)=j|X(m)=i),$$ where $i, j \in S$, $n \geq 1$, and $m \geq 0$. Let $d(i)$ be the period of state $i \in S$, i.e., $$d(i):=\gcd\{n \geq 1: p_{ii}(n)>0 \}.$$

I want to prove that $d(i)=d(j)$ for two states $i, j \in S$ which intercommunicate.

I know how to prove that two states in the same communicating class of a Markov chain have the same period (see Show that two states in the same communicating class of a Markov chain must have the same period), but the fact that $X(n)_{n \geq 0}$ is homogeneous is not used, so I am not sure how to prove this. Thank you for your help.

AMM
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1 Answers1

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The (time) homogeneity of Markov chains is often assumed as part of the definition of Markov chain. This is the case in the link you referenced.

The concepts of communicating classes and periods really wouldn't make sense for a non-homogeneous Markov chain where transition probabilities might be $0$ at some times and nonzero at others.

Robert Israel
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