I am studying MERW and for some reasons, i would like to know if, if I have $(X_n)$ an irreducible Markov Chain, I can say that $\mathbb{P}(X_1=x | X_0=a, X_n = b)$ goes to $\mathbb{P}(X_1=x | X_0=a)$ when $n$ goes to $+\infty$. It would be very helpful and I find it intuitive but I don't know how to prove it, or even if it's true...
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4$\begingroup$ If the Markov chain has a finite number of states, then you can simply write down both expressions in terms of the transition matrix and check that this convergence indeed holds. If you have infinitely many states, then I suspect that it isn't true in general, unless you also assume positive recurrence or something like that (it probably holds under somewhat weaker assumptions, I suspect that having a trivial Martin boundary is enough). $\endgroup$– Martin HairerCommented Oct 8 at 19:24
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