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Let $(X_n)$ be an irreducible Markov chain on a countable state space $S$. Suppose there's an non-empty set $A\subseteq S$ such that for some $x\notin A$

$$ P_x(\tau_A<\infty)<1$$

Where $\tau_A :=\inf\{n\geqslant 1: X_n\in A\}$ is the hitting time of A.

Can you assure that $(X_n)$ is transient ?

So far the definition asks for some state $z\in S$ such that $P_z(\tau_z<\infty)<1$ (which implies that this last condition holds for every state).

I've come across this problem while checking the proof of Proposition 1.3 of this notes from Hairer.

Max
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1 Answers1

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Yes. Let $a \in A$. Since the chain is irreducible, there exists $n \in \mathbb{N}$ such that $P_{a}(X_{n}=x,X_{1}^{n-1} \neq a) > 0$. Therefore, \begin{align*} P_{a}(\tau_{a}=\infty) &\geq P_{a}(X_{1}^{n-1} \neq a,X_{n}=x,\tau_{a}=\infty) \\ &\geq P_{a}(X_{n}=x,X_{1}^{n-1} \neq a)P_{x}(\tau_{a}=\infty) \\ &\geq P_{a}(X_{n}=x,X_{1}^{n-1} \neq a)P_{x}(\tau_{A}=\infty) > 0 \end{align*} Therefore, $a$ is transient. Since the chain is irreducible, it is transient.

madprob
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