Thread: Markov Chain

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

$\displaystyle \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.

This is the one step ahead property.. I think you just have to show an inductive proof.