# Thread: 2-Periodic Markov Chain

1. ## 2-Periodic Markov Chain

Could you help me with this question please?

If a is a state of a Markov chain, we say that a is 2-periodic if $p_{aa}^{k}=0$ for all odd k.
a) Show that if i is 2-periodic and j is a element of the same communicating class as i then j is 2-periodic. (We say that 2-periodicity is a class property.)
[Hint: Assume that i is 2-periodic but j is not 2-periodic and deduce a contradiction.]
Thank you.

2. Suppose that i is 2-periodic and there exists an odd n such that $p_{jj}^{n}>0$. Now there exists an m, k such that $p_{ij}^m>0$ and $p_{ji}^k>0$. Now one way you can return to i from i is via $p_{i i}^{n+m+k} \geq p_{ij}^m p_{j j}^n p_{j i}^k>0$.

Is m+k odd or even (remember that n is odd and n+k+m must also be odd)? What does that tell you about $p_{i i}^{m+k} \geq p_{ij}^m p_{ji}^k >0$?