# Irreducible Markov chain rotating cyclically

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?

• 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. – amd Apr 25 at 6:15
• Thank you. Do you have any idea how should I start thinking about proving it? – hesse Apr 26 at 1:10

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 $$j=d$$.