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State \(i\) communicates with state \(j\) (or \(j\) is accessible from \(i\)), denoted \(i \to j\) if \(p_{i,j}(n)>0\) for some \(n\geq 0\) that is, there is positive probability for the chain to visit \(j\) when starting from \(i\).
- If \(i \neq j\) this is equivalent to \(f_{i,j}=\mathbb{P}(T_{j}<\infty|X_{0}=i)>0\).
- If \(i = j\), then \(p_{i,i}(0)=1\) so that \(i \to i\) but it is possible for \(f_{i,i}=\sum_{n=1}^\infty f_{i,i}(n)=0\), that is, the chain might leave \(i\) immediately and never return to it.
States \(i\) and \(j\) inter-communicate if \(i \to j\) and \(j \to i\), denoted \(i\leftrightarrow j\) and we have the following results:
- States \(i\) and \(j\) have the same period;
- State \(i\) is transient iff \(j\) is transient; and
- State \(i\) is null persistent iff \(j\) is.
A set of states \(C\) is irreducible if for all \(i,j\in C\), \(i\leftrightarrow j\), that is all states within \(C\) inter-communicate.
A set of states \(C\) is closed if for all \(i \in C\), \(p_{i,j}=0\) for all \(j\not\in C\), that is, the chain never leaves \(C\) once it has entered. Not that clearly the set consisting of one absorbing state is closed.