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I am going through Knowing the Odds by John B. Walsh, and I am stuck at one of the exercises there (which is important to understand some next theorem).

The exercise is as follows:

Let $X$ be an irreducible chain with period $d$. Show that there exist $d$ disjoint sets of states, $S_1,\dots,S_d$, such that if the chain starts in $S_1$, it rotates through $S_2, S_3, \dots ,S_d$ and then back to $S_1$.

I don't even understand what he exactly means by "rotation" and do not know where to start. Any thoughts? Where do I start the proof?

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  • $\begingroup$ He he means that from an element of $S_1$ $X$ transitions to an element of $S_2$, then to an element of $S_3$, &c, until it gets to an element of $S_d$, whence the next transition takes it back to some element of $S_1$, though not necessarily the same one that it started at. $\endgroup$ – amd Apr 25 at 6:15
  • $\begingroup$ Thank you. Do you have any idea how should I start thinking about proving it? $\endgroup$ – hesse Apr 26 at 1:10
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Walsh means that the system cycles through these classes: all of the transitions out of states in $S_1$ are to states in $S_2$, all of the transitions out of states in $S_2$ are to states in $S_3$, and so on, with all of the transitions out of states in $S_d$ going to states in $S_1$.

One way to approach a proof is to define the state sets $$S_m = \{j \mid P_{1j}^{nd+m}\gt0 \text{ for some } n\ge0\}, 1\le m\le d.$$ You can then use path lengths and the properties of modular arithmetic to argue that each $j$ appears in exactly one of these sets. You can then use similar ideas to argue that if $j\in S_m$ and $P_{jk}\gt0$, that $k = j+1$ when $j\lt d$ and $k=1$ when $j=d$.

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