Markov Chain

**harbottle** · Nov 2nd 2010, 09:11 PM

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.

**Scopur** · Nov 5th 2010, 08:30 AM

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

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