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Consider $(X_n)$ a discrete-time Markov chain with values in a countable space $E$. Let $x$ be a transient state, let $R$ denote the set of recurrent states and $P_x$ the probability measure when starting at $x$. Define $T=\inf\{n\geq 0, X_n \in R\}$.

Is it true that either $P_x(T=\infty)=1$ or $P_x(T<\infty)=1$ ?

I believe this has been claimed in some lecture notes I'm reading. However, no proof is given.

I'd consider two cases: whether $x$ communicates with some recurrent $y$ or not. If there is no such $y$, then $T$ is always $\infty$. On the contrary, if $x$ communicates with some recurrent $y$, I don't think $y$ is necessarily hit with probability $1$, thus there is no reason to believe that $P_x(T<\infty)=1$.

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Consider a random walk on $\Bbb N_0$ with transition probabilities as follows. $p_{o,o}=1,p_{1,0}=1/2=p_{1,2},p_{n,n+1}=1$ for all $n>1$. Then $0$ is the only recurrent state, and the hitting time of $0$ starting from $1$ is $1$ with probability $1/2$ and $\infty$ with probability $1/2$.

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