Given any positive integer, how can I think of a Markov Chain (states and transition probabilities) to have that integer as the period of two of its states?
Thanks.
Given any positive integer, how can I think of a Markov Chain (states and transition probabilities) to have that integer as the period of two of its states?
Thanks.
To be truly stochastic, assume that the state space of the Markov chain is $S\times C_d$ where $C_d=\mathbb{Z}/d\mathbb{Z}$ is the discrete circle of size $d$. And assume that the only transitions with positive probabilities go from some state $(x,k)$ to some state $(y,k+1)$ with $x$ and $y$ in $S$ and $k$ in $C_d$. Then the period of this Markov chain is a multiple of $d$. To ensure that the period is exactly $d$, assume further that there exists an $S$-valued sequence $(x_k)$ indexed by $C_d$ (hence $x_{k+d}=x_k$ for every integer $k$) such that the transition from $(x_k,k)$ to $(x_{k+1},k+1)$ has positive probability, for every $k$ in $C_d$. (Note that every Markov chain with period $d$ may be encoded like this.)
Hint: No probabilities are needed. Just put them on a (directed) cycle.