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We have a discrete time Markov chain taking discrete values $\{X_n, n\geq 0\}$. The question is that whether $P(X_n=m| X_{n-1}\neq r, \dots, X_1\neq r, X_0=i) \ \mbox{ for } \ i,m\neq r$ is related to another Markov chain, let us say, $Y_n$ where the state $r$ is excluded.

Notation: Call $p_{ij} := P(X_{n+1}=j| X_n=i)$ and $q_{ij} := P(Y_{n+1}=j|Y_n=i)$

What I don't understand is that, if $X_n$ is a Markov chain, then $P(X_{n+1}=j|X_0=i)=P(X_{n+1}=j)$ isn't it? I don't see the contribution of knowing $X_0=i$ :( Because I've seen the notation: $p_{ij}^{(n)}=P(X_n=j| X_0=i)$ but I don't understand it.

Thanks very much for your time and help.

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It is not true that $P(X_{n+1} = j\ |\ X_0 = i) = P(X_{n+1} = j)$. (I mean, not true in general.) – Tunococ Jan 22 '13 at 8:55

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