Definition 3.6.1. Recurrent and transient states Let $\left(X_n\right)_{n \geq 0}$ be a Markov chain with the state space $S$. A state $i \in S$ is said to be recurrent if $f_i=\mathbb{P}\left(T_i<\infty \mid X_0=i\right)=1$. A state $i \in S$ is said to be transient if $f_i=\mathbb{P}\left(T_i<\infty \mid X_0=i\right)<1$.