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Let $$f_{kk}^{(n)}:=\Pr(X_n=k,X_v\ne k,1\le v\le n-1\mid X_0=k),~n\in \mathbb Z^+ .$$ Attributing to the comments of @Zhanxiong , I have added the other two cases to the Case 1.

Case 1.

Is there a discrete time-homogeneous Markov chain $(X_n)_{n \geq 0}$ in which one transient state $i$ satisfies $ \sum_{n=1}^{\infty}nf^{(n)}_{ii}<\infty $?

Case 2.

Is there a discrete time-homogeneous Markov chain $(X_n)_{n \geq 0}$ in which one transient state $j$ satisfies $ \sum_{n=1}^{\infty}nf^{(n)}_{jj}=\infty $?

Case 3.

Is there a discrete time-homogeneous Markov chain $(X_n)_{n \geq 0}$ in which two distinct transient states $i$ and $j$, $i$ satisfies $ \sum_{n=1}^{\infty}nf^{(n)}_{ii}<\infty $ and $j$ satisfies $\sum_{n=1}^{\infty}nf^{(n)}_{jj}=\infty $?

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  • $\begingroup$ I refer you to study Example 8.13 in Probability and Measure (3rd ed.) by Patrick Billingsley. $\endgroup$
    – Zhanxiong
    Oct 17, 2023 at 3:47

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