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Math Help - Markov Chain

  1. #1
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    Markov Chain

    For a Markov chain \{X_n, n\geq 0 \}, show that

    \mathbf{P}(X_k=i_k|X_j=i_j,\;\text{for all}\; j\neq k) = \mathbf{P}(X_k=i_k|X_{k-1}=i_{k-1},X_{k+1}=i_{k+1})

    I know what the Markov property is, but I am worthless at manipulating contidional probabilities like this one. Any place to start would be helpful.

    I write down the definition of conditional probability, and no matter which way I try to manipulate it I end up with a gigantic mess. Any help would be appreciated.
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  2. #2
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    This is the one step ahead property.. I think you just have to show an inductive proof.
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