Definition (Communicates)

Let $X_{n},n\ge0$ be a MC, with transition matrix $P$ and the state space $S$. Let $i,j\in S$ be two states. We say $i$ communicates with $j$ if $$i\to j \mbox{ and }j\to i$$ or “$i$ and $j$ lead to each other” and write $$i\leftrightarrow j$$

Definition (Communicating Class)

Based on , we can partition the state space $S$ into communication classes, $\mathcal{C}_{i}, \ i\in\mathbb{N}$ where:

1. $S=\cup^{\infty}_{n=1}\mathcal{C}_{n}$, where $\mathcal{C}_{n}$’s are disjoint.
2. For $a_{1},a_{2}\in\mathcal{C}_{n}\implies \ a_{1}\leftrightarrow a_{2}$ or “any two elements in the same class communicate”
3. For $b\in\mathcal{C}_{n}, \ c\in\mathcal{C}_{m}, \ n\not=m\implies b\not\leftrightarrow c$ or “any two elements in different classes don’t communicate”

Definition (Closed communicating class)

A $\mathcal{C}$ is closed if $$p_{ij}=0, \ \mbox{ for any }i\in\mathcal{C}, \ j\not\in\mathcal{C}$$Thus once a MC enters $\mathcal{C}$, there is not escape.