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The following theorem is from Marikov Chains by Norris:

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I don't understand the "In particular" part. In the first limit, $P(X_n=j)$ depends not only on the transition probability matrix $P$ but also the initial distribution $\lambda$. But in the second limit, the $n$-step transition probability $p_{ij}^{n}$ only depends on the transition probability matrix $P$. How would the second one be a particular case of the first one?

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    After enough time, the initial distribution doesn't matter. – 2017-02-26
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    Note that $p_{ij}^n $ is just the same as $P(X_n=j)$ where the initial distribution is $\delta_i$. – 2017-02-26
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    @Shalop: Stupid me. I didn't notice that this is rather trivial, thanks. If you could write your comment as an answer, I would accept it. – 2017-02-26

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