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Status quo:
We consider a irreducible, aperiodic Markov chain $(X_n)_{n\in\mathbb{N}}$ on a countable set $S$ with tranistion function $p(\cdot,\cdot)$.

Now we want to examine $\underset{n\rightarrow\infty}{\text{lim}}~p^n(i,j)$ for arbitrary $i,j\in S$.

If $j$ is transient we can show that $\sum_{n=0}^\infty p^n(i,j)<\infty$ and therefore we know that $\underset{n\rightarrow\infty}{\text{lim}}~p^n(i,j)=0$. If $j$ is positive recurrent we can show by nice coupling argument (see for example Durrett) that $\underset{n\rightarrow\infty}{\text{lim}}~p^n(i,j)=\pi(j)$, where $\pi$ is the unique stationary distribution of the chain.

The question:
My concern is the null recurrent case. I looked through a bunch of textbooks and I could not find a proof for the fact that $\underset{n\rightarrow\infty}{\text{lim}}~p^n(i,j)=0$ in this case. Most textbooks mention it but none of them gives a proof and I can't proof it either.
Any ideas, thoughts and especially references are highly appreciated. Thanks!

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    $\begingroup$ ?? Huh? Which textbooks did you check? $\endgroup$ – Did Mar 12 '15 at 17:46
  • $\begingroup$ @Did The usual suspects (Durrett, Klenke, Levin, lecturenotes) - maybe I was blind. But now I found it (see my answer). $\endgroup$ – MathProb Mar 12 '15 at 20:48
  • $\begingroup$ @Did Another thing since you have always been a help with my probability questions. Any thoughts on this: math.stackexchange.com/questions/1184926/… ? I would be very grateful. $\endgroup$ – MathProb Mar 12 '15 at 21:43
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After some more research I found a nice answer in the book by Norris Theorem 1.8.5. It also makes use of a Coupling argument with the choice of a shifted measure.

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