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

where $$f_{ii}^{(n)}:=\Pr(X_n=i,X_v\ne i,1\le v\le n-1\mid X_0=i),n\in \mathbb Z^+\quad .$$

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  • $\begingroup$ A more interesting question would be is there a transient state $i$ satisfies $\sum_{n = 1}^\infty nf_{ii}^{(n)} = \infty$? $\endgroup$
    – Zhanxiong
    Oct 16, 2023 at 18:17
  • $\begingroup$ It's better to ask a new question given an answer to your old question has already been posted. $\endgroup$
    – Zhanxiong
    Oct 17, 2023 at 1:38

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Consider a 2-state $\{0, 1\}$ Markov chain with transition matrix $P = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ 0 & 1 \end{bmatrix}$. It is easy to verify that $f_{00}^{(1)} = \frac{1}{2}$ and $f_{00}^{(n)} = 0$ for $n \geq 2$ (because $1$ is an absorbing state). Clearly state $0$ is transient and $$\sum_{n = 1}^\infty nf_{00}^{(n)} = \frac{1}{2} < \infty.$$

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