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Aperiodicity of a Markov Chain

  1. Apr 11, 2008 #1
    1. The problem statement, all variables and given/known data

    Transition matrix is

    0 0 1
    0 0 1
    (1/3) (2/3) 0

    "argue that this chain is aperiodic"

    2. Relevant equations

    definition of aperiodicity - there must exist a time n such that there is a non-zero probability of going from state i to state j for all i & j

    3. The attempt at a solution

    This definition doesn't seem to hold for my chain ... for example, to go from state 1 to state 2 n has to be odd.. but to go from state 1 to state 1 or 3 n has to be even..

    Am I just getting this definition muddled up? Could someone elaborate on it for me? Thanks
  2. jcsd
  3. Apr 12, 2008 #2
  4. Apr 13, 2008 #3
    The chain is aperiodic 1->3->2->3->1
    You can get from any position to any other (it doesn't have to be in one step..)
  5. Apr 13, 2008 #4
    Yeah, I can see it's not periodic and hence must be apeiodic, but what's going on with that definition? My understanding of it is that there has to be a special (fixed) value of n where you can go from any one state to all the others, including back to that state... but that doesn't seem to hold here... thanks for replying
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