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Math Help - Markov Chains + Transient States Proof - Help

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    Markov Chains + Transient States Proof - Help

    Hi all,

    I am trying to proof that:

    P^n(kj) = 0 for all n when K is recurrent, where P(kj) is the expected # of time periods that a markov chain is in state j given that it starts in state k.

    I am totally lost and don't even know where to start. Please demonstrate how to tackle this proof. Thanks.

    EDIT:
    Ok, I have figured out the following:

    P(kj) = 0 + Sum(P[1st transition is in state k) * E[# of visits to j | starts at k]

    So I KNOW based on this equation why it will turn out to zero. Obviously (Sum(P[1st transition is in state k) * E[# of visits to j | starts at k]) has to = 0 for the proof to work, but how do I figure that part out?????

    Can somebody show me why:

    Sum(P[1st transition is in state k) * E[# of visits to j | starts at k] = 0 ? Thanks
    Last edited by spearfish; November 7th 2010 at 09:14 AM.
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