class structure

Let ${(X_n)}_{n\ge 1}$ be a stationary Markov chain and let $i$ and $j$ be states in the indexing set. We say that $i$ leads to $j$ or $j$ is accessible from $i$, and write $i\to j$, if it is possible for the chain to get from state $i$ to state $j$:

$$i\to j\iff P(X_n=j:X_0=i)>0\mathit{\hspace{1em}}\text{for some}\mathit{\hspace{1em}}n\ge 0$$

If $i\to j$ and $j\to i$ we say $i$ communicates with $j$ and write $i\leftrightarrow j$. $\leftrightarrow$ is an equivalence relation (easy to prove). The equivalence classes of this relation are the communicating classes of the chain. If there is just one class, we say the chain is an irreducible chain.

A class $C$ is a closed class if $i\in C$ and $i\to j$ implies that $j\in C$ “Once the chain enters a closed class, it cannot leave it”

A state $i$ is an absorbing state if $\{i\}$ is a closed class.