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Math Help - 2-Periodic Markov Chain

  1. #1
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    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.
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  2. #2
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    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?
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