Lemma (Equivalent Definitions of Transience)

1. Starting from state $i$, w.p.1., it will visit $i$ finitely many times, $$P_{i}(X_{n}=i \ \ i.o.)=0$$
2. $$\sum\limits_{n=0}^{\infty}(P^{n})_{ii}<\infty$$
3. The occupation time for any state $\alpha\in X$ is finite $$E_{\alpha}[\eta_{\alpha}]<\infty$$