Does someone have an example for a discrete time (time-homogeneous) Markov chain in a three-dimensional state space that is characterized by a transition matrix resulting in periodic behaviour?

I am aware of the following (unique) example

  • in two-dimensional state space: $\hspace{2em} P=\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$ and

  • its analog in three-dimensional state space $P=\begin{pmatrix} 0 & 1& 0\\ 1 & 0& 0\\ 0 & 0& 1\\ \end{pmatrix}$.

Is there an intuitive way of constructing such transition matrices?


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